Analysis
8.2. Summation on infinite sets

Imports

import Mathlib.Tactic
import Analysis.Section_7_2
import Analysis.Section_7_3
import Analysis.Section_7_4
import Analysis.Section_8_1

Analysis I, Section 8.2: Summation on infinite sets

I have attempted to make the translation as faithful a paraphrasing as possible of the original text. When there is a choice between a more idiomatic Lean solution and a more faithful translation, I have generally chosen the latter. In particular, there will be places where the Lean code could be "golfed" to be more elegant and idiomatic, but I have consciously avoided doing so.

Main constructions and results of this section:

Absolute convergence and summation on countably infinite or general sets.
Connections with Mathlib's Summable and tsum.
The Riemann rearrangement theorem.

Some non-trivial API is provided beyond what is given in the textbook in order connect these notions with existing summation notions.

After this section, the summation notation developed here will be deprecated in favor of Mathlib's API for Summable and tsum.

namespace Chapter8open Chapter7 Chapter7.Series Finset Function Filter

Definition 8.2.1 (Series on countable sets). Note that with this definition, functions defined on finite sets will not be absolutely convergent; one should use AbsConvergent' instead for such cases.

abbrev AbsConvergent {X:Type} (f: X → ℝ) : Prop := ∃ g: ℕ → X, Bijective g ∧ (f ∘ g: Series).absConverges

theorem AbsConvergent.mk {X: Type} {f:X → ℝ} {g:ℕ → X} (h: Bijective g) (hfg: (f ∘ g:Series).absConverges) : AbsConvergent f := byX:Typef:X → ℝg:ℕ → Xh:Bijective ghfg:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges⊢ AbsConvergent f use gAll goals completed! 🐙

The definition has been chosen to give a sensible value when X is finite, even though AbsConvergent is by definition false in this context.

open Classical in

noncomputable abbrev Sum {X:Type} (f: X → ℝ) : ℝ := if h: AbsConvergent f then (f ∘ h.choose:Series).sum else
  if _hX: Finite X then (∑ x ∈ @univ X (Fintype.ofFinite X), f x) else 0

              theorem Sum.of_finite {X:Type} [hX:Finite X] (f:X → ℝ) : Sum f = ∑ x ∈ @Finset.univ X (Fintype.ofFinite X), f x := byX:TypehX:Finite Xf:X → ℝ⊢ Sum f = ∑ x, f x
  have : ¬ AbsConvergent f := byX:TypehX:Finite Xf:X → ℝ⊢ Sum f = ∑ x, f x
    by_contra!X:TypehX:Finite Xf:X → ℝthis:AbsConvergent f⊢ False; choose g hg _ using thisX:TypehX:Finite Xf:X → ℝg:ℕ → Xhg:Bijective ga✝:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges⊢ False
    rw [←hg.finite_iff, ←not_infinite_iff_finite] at hXX:TypehX:¬Infinite ℕf:X → ℝg:ℕ → Xhg:Bijective ga✝:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges⊢ False; apply hXX:TypehX:¬Infinite ℕf:X → ℝg:ℕ → Xhg:Bijective ga✝:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges⊢ Infinite ℕ; infer_instanceAll goals completed! 🐙
  simp [Sum, this, hX]All goals completed! 🐙theorem AbsConvergent.comp {X: Type} {f:X → ℝ} {g:ℕ → X} (h: Bijective g) (hf: AbsConvergent f) : (f ∘ g:Series).absConverges := byX:Typef:X → ℝg:ℕ → Xh:Bijective ghf:AbsConvergent f⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges
  choose g' hbij hconv using hfX:Typef:X → ℝg:ℕ → Xh:Bijective gg':ℕ → Xhbij:Bijective g'hconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConverges⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges
  choose g'_inv hleft hright using bijective_iff_has_inverse.mp hbijX:Typef:X → ℝg:ℕ → Xh:Bijective gg':ℕ → Xhbij:Bijective g'hconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg'_inv:X → ℕhleft:LeftInverse g'_inv g'hright:RightInverse g'_inv g'⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges
  have hG : Bijective (g'_inv ∘ g) := .comp ⟨hright.injective, hleft.surjective⟩ hX:Typef:X → ℝg:ℕ → Xh:Bijective gg':ℕ → Xhbij:Bijective g'hconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg'_inv:X → ℕhleft:LeftInverse g'_inv g'hright:RightInverse g'_inv g'hG:Bijective (g'_inv ∘ g)⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges
  convert (absConverges_of_permute hconv hG).1 using 4 with nh.e'_1.h.e'_2.h.h₂X:Typef:X → ℝg:ℕ → Xh:Bijective gg':ℕ → Xhbij:Bijective g'hconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg'_inv:X → ℕhleft:LeftInverse g'_inv g'hright:RightInverse g'_inv g'hG:Bijective (g'_inv ∘ g)n:ℤa✝:n ≥ 0⊢ (f ∘ g) n.toNat = (fun n ↦ (f ∘ g') ((g'_inv ∘ g) n)) n.toNat
  simp [hright (g n.toNat)]All goals completed! 🐙theorem Sum.eq {X: Type} {f:X → ℝ} {g:ℕ → X} (h: Bijective g) (hfg: (f ∘ g:Series).absConverges) : (f ∘ g:Series).convergesTo (Sum f) := byX:Typef:X → ℝg:ℕ → Xh:Bijective ghfg:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum f)
  have : AbsConvergent f := .mk h hfgX:Typef:X → ℝg:ℕ → Xh:Bijective ghfg:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesthis:AbsConvergent f⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum f)
  simp [Sum, this]X:Typef:X → ℝg:ℕ → Xh:Bijective ghfg:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesthis:AbsConvergent f⊢ { m := 0, seq := fun n ↦ if 0 ≤ n then f (g n.toNat) else 0, vanish := ⋯ }.convergesTo
  { m := 0, seq := fun n ↦ if 0 ≤ n then f (⋯.choose n.toNat) else 0, vanish := ⋯ }.sum
  choose hbij hconv using this.choose_specX:Typef:X → ℝg:ℕ → Xh:Bijective ghfg:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesthis:AbsConvergent fhbij:Bijective (Exists.choose this)hconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ Exists.choose this) n.toNat else 0, vanish := ⋯ }.absConverges⊢ { m := 0, seq := fun n ↦ if 0 ≤ n then f (g n.toNat) else 0, vanish := ⋯ }.convergesTo
  { m := 0, seq := fun n ↦ if 0 ≤ n then f (⋯.choose n.toNat) else 0, vanish := ⋯ }.sum
  set g' := this.chooseX:Typef:X → ℝg:ℕ → Xh:Bijective ghfg:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesthis:AbsConvergent fg':ℕ → X := Exists.choose thishbij:Bijective g'hconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConverges⊢ { m := 0, seq := fun n ↦ if 0 ≤ n then f (g n.toNat) else 0, vanish := ⋯ }.convergesTo
  { m := 0, seq := fun n ↦ if 0 ≤ n then f (⋯.choose n.toNat) else 0, vanish := ⋯ }.sum
  choose g'_inv hleft hright using bijective_iff_has_inverse.mp hbijX:Typef:X → ℝg:ℕ → Xh:Bijective ghfg:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesthis:AbsConvergent fg':ℕ → X := Exists.choose thishbij:Bijective g'hconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg'_inv:X → ℕhleft:LeftInverse g'_inv g'hright:RightInverse g'_inv g'⊢ { m := 0, seq := fun n ↦ if 0 ≤ n then f (g n.toNat) else 0, vanish := ⋯ }.convergesTo
  { m := 0, seq := fun n ↦ if 0 ≤ n then f (⋯.choose n.toNat) else 0, vanish := ⋯ }.sum
  convert convergesTo_sum (converges_of_absConverges hfg) using 1h.e'_2X:Typef:X → ℝg:ℕ → Xh:Bijective ghfg:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesthis:AbsConvergent fg':ℕ → X := Exists.choose thishbij:Bijective g'hconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg'_inv:X → ℕhleft:LeftInverse g'_inv g'hright:RightInverse g'_inv g'⊢ { m := 0, seq := fun n ↦ if 0 ≤ n then f (⋯.choose n.toNat) else 0, vanish := ⋯ }.sum =
  { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.sum
  have hG : Bijective (g'_inv ∘ g) := .comp ⟨hright.injective, hleft.surjective⟩ hh.e'_2X:Typef:X → ℝg:ℕ → Xh:Bijective ghfg:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesthis:AbsConvergent fg':ℕ → X := Exists.choose thishbij:Bijective g'hconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg'_inv:X → ℕhleft:LeftInverse g'_inv g'hright:RightInverse g'_inv g'hG:Bijective (g'_inv ∘ g)⊢ { m := 0, seq := fun n ↦ if 0 ≤ n then f (⋯.choose n.toNat) else 0, vanish := ⋯ }.sum =
  { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.sum
  convert (absConverges_of_permute hconv hG).2 using 4 with _ nh.e'_3.h.e'_1.h.e'_2.hX:Typef:X → ℝg:ℕ → Xh:Bijective ghfg:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesthis:AbsConvergent fg':ℕ → X := Exists.choose thishbij:Bijective g'hconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg'_inv:X → ℕhleft:LeftInverse g'_inv g'hright:RightInverse g'_inv g'hG:Bijective (g'_inv ∘ g)n:ℤ⊢ (if n ≥ 0 then (f ∘ g) n.toNat else 0) = if n ≥ 0 then (fun n ↦ (f ∘ g') ((g'_inv ∘ g) n)) n.toNat else 0
  by_cases hn : n ≥ 0posX:Typef:X → ℝg:ℕ → Xh:Bijective ghfg:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesthis:AbsConvergent fg':ℕ → X := Exists.choose thishbij:Bijective g'hconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg'_inv:X → ℕhleft:LeftInverse g'_inv g'hright:RightInverse g'_inv g'hG:Bijective (g'_inv ∘ g)n:ℤhn:n ≥ 0⊢ (if n ≥ 0 then (f ∘ g) n.toNat else 0) = if n ≥ 0 then (fun n ↦ (f ∘ g') ((g'_inv ∘ g) n)) n.toNat else 0negX:Typef:X → ℝg:ℕ → Xh:Bijective ghfg:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesthis:AbsConvergent fg':ℕ → X := Exists.choose thishbij:Bijective g'hconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg'_inv:X → ℕhleft:LeftInverse g'_inv g'hright:RightInverse g'_inv g'hG:Bijective (g'_inv ∘ g)n:ℤhn:¬n ≥ 0⊢ (if n ≥ 0 then (f ∘ g) n.toNat else 0) = if n ≥ 0 then (fun n ↦ (f ∘ g') ((g'_inv ∘ g) n)) n.toNat else 0 <;>posX:Typef:X → ℝg:ℕ → Xh:Bijective ghfg:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesthis:AbsConvergent fg':ℕ → X := Exists.choose thishbij:Bijective g'hconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg'_inv:X → ℕhleft:LeftInverse g'_inv g'hright:RightInverse g'_inv g'hG:Bijective (g'_inv ∘ g)n:ℤhn:n ≥ 0⊢ (if n ≥ 0 then (f ∘ g) n.toNat else 0) = if n ≥ 0 then (fun n ↦ (f ∘ g') ((g'_inv ∘ g) n)) n.toNat else 0negX:Typef:X → ℝg:ℕ → Xh:Bijective ghfg:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesthis:AbsConvergent fg':ℕ → X := Exists.choose thishbij:Bijective g'hconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg'_inv:X → ℕhleft:LeftInverse g'_inv g'hright:RightInverse g'_inv g'hG:Bijective (g'_inv ∘ g)n:ℤhn:¬n ≥ 0⊢ (if n ≥ 0 then (f ∘ g) n.toNat else 0) = if n ≥ 0 then (fun n ↦ (f ∘ g') ((g'_inv ∘ g) n)) n.toNat else 0 simp [hn, hright (g n.toNat)]All goals completed! 🐙theorem Sum.of_comp {X Y:Type} {f:X → ℝ} (h: AbsConvergent f) {g: Y → X} (hbij: Bijective g) : AbsConvergent (f ∘ g) ∧ Sum f = Sum (f ∘ g) := byX:TypeY:Typef:X → ℝh:AbsConvergent fg:Y → Xhbij:Bijective g⊢ AbsConvergent (f ∘ g) ∧ Sum f = Sum (f ∘ g)
  choose g' hbij' hconv' using hX:TypeY:Typef:X → ℝg:Y → Xhbij:Bijective gg':ℕ → Xhbij':Bijective g'hconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConverges⊢ AbsConvergent (f ∘ g) ∧ Sum f = Sum (f ∘ g)
  choose g_inv hleft hright using bijective_iff_has_inverse.mp hbijX:TypeY:Typef:X → ℝg:Y → Xhbij:Bijective gg':ℕ → Xhbij':Bijective g'hconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg_inv:X → Yhleft:LeftInverse g_inv ghright:RightInverse g_inv g⊢ AbsConvergent (f ∘ g) ∧ Sum f = Sum (f ∘ g)
  have hbij_g_inv_g' : Bijective (g_inv ∘ g') := .comp ⟨hright.injective, hleft.surjective⟩ hbij'X:TypeY:Typef:X → ℝg:Y → Xhbij:Bijective gg':ℕ → Xhbij':Bijective g'hconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg_inv:X → Yhleft:LeftInverse g_inv ghright:RightInverse g_inv ghbij_g_inv_g':Bijective (g_inv ∘ g')⊢ AbsConvergent (f ∘ g) ∧ Sum f = Sum (f ∘ g)
  have hident : (f ∘ g) ∘ g_inv ∘ g' = f ∘ g' := byX:TypeY:Typef:X → ℝh:AbsConvergent fg:Y → Xhbij:Bijective g⊢ AbsConvergent (f ∘ g) ∧ Sum f = Sum (f ∘ g) ext nhX:TypeY:Typef:X → ℝg:Y → Xhbij:Bijective gg':ℕ → Xhbij':Bijective g'hconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg_inv:X → Yhleft:LeftInverse g_inv ghright:RightInverse g_inv ghbij_g_inv_g':Bijective (g_inv ∘ g')n:ℕ⊢ ((f ∘ g) ∘ g_inv ∘ g') n = (f ∘ g') n; simp [hright (g' n)]All goals completed! 🐙
  refine ⟨ ⟨ g_inv ∘ g', ⟨ hbij_g_inv_g', byX:TypeY:Typef:X → ℝg:Y → Xhbij:Bijective gg':ℕ → Xhbij':Bijective g'hconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg_inv:X → Yhleft:LeftInverse g_inv ghright:RightInverse g_inv ghbij_g_inv_g':Bijective (g_inv ∘ g')hident:(f ∘ g) ∘ g_inv ∘ g' = f ∘ g'⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then ((f ∘ g) ∘ g_inv ∘ g') n.toNat else 0, vanish := ⋯ }.absConverges convert hconv'All goals completed! 🐙 ⟩ ⟩, ?_ ⟩
  have h := eq (f := f ∘ g) hbij_g_inv_g' (byX:TypeY:Typef:X → ℝg:Y → Xhbij:Bijective gg':ℕ → Xhbij':Bijective g'hconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg_inv:X → Yhleft:LeftInverse g_inv ghright:RightInverse g_inv ghbij_g_inv_g':Bijective (g_inv ∘ g')hident:(f ∘ g) ∘ g_inv ∘ g' = f ∘ g'⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then ((f ∘ g) ∘ g_inv ∘ g') n.toNat else 0, vanish := ⋯ }.absConverges convert hconv'All goals completed! 🐙)
  rw [hident] at hX:TypeY:Typef:X → ℝg:Y → Xhbij:Bijective gg':ℕ → Xhbij':Bijective g'hconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.absConvergesg_inv:X → Yhleft:LeftInverse g_inv ghright:RightInverse g_inv ghbij_g_inv_g':Bijective (g_inv ∘ g')hident:(f ∘ g) ∘ g_inv ∘ g' = f ∘ g'h:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g') n.toNat else 0, vanish := ⋯ }.convergesTo (Sum (f ∘ g))⊢ Sum f = Sum (f ∘ g)
  solve_by_elim [convergesTo_uniq, eq]All goals completed! 🐙@[simp]
theorem Finset.Icc_eq_cast (N:ℕ) : Icc 0 (N:ℤ) = map Nat.castEmbedding (.Icc 0 N) := byN:ℕ⊢ Icc 0 ↑N = Finset.map Nat.castEmbedding (Icc 0 N)
  ext nhN:ℕn:ℤ⊢ n ∈ Icc 0 ↑N ↔ n ∈ Finset.map Nat.castEmbedding (Icc 0 N); simphN:ℕn:ℤ⊢ 0 ≤ n ∧ n ≤ ↑N ↔ ∃ a ≤ N, ↑a = n; constructorh.mpN:ℕn:ℤ⊢ 0 ≤ n ∧ n ≤ ↑N → ∃ a ≤ N, ↑a = nh.mprN:ℕn:ℤ⊢ (∃ a ≤ N, ↑a = n) → 0 ≤ n ∧ n ≤ ↑N
  .h.mpN:ℕn:ℤ⊢ 0 ≤ n ∧ n ≤ ↑N → ∃ a ≤ N, ↑a = n intro ⟨ hn, _ ⟩h.mpN:ℕn:ℤhn:0 ≤ nright✝:n ≤ ↑N⊢ ∃ a ≤ N, ↑a = n; lift n to ℕ using hnh.mpN:ℕn:ℕright✝:↑n ≤ ↑N⊢ ∃ a ≤ N, ↑a = ↑n; use nhN:ℕn:ℕright✝:↑n ≤ ↑N⊢ n ≤ N ∧ ↑n = ↑n; simp_allAll goals completed! 🐙
  rintro ⟨ _, ⟨ _, rfl ⟩ ⟩h.mprN:ℕw✝:ℕleft✝:w✝ ≤ N⊢ 0 ≤ ↑w✝ ∧ ↑w✝ ≤ ↑N; simp_allAll goals completed! 🐙theorem Finset.Icc_empty {N:ℤ} (h: ¬ N ≥ 0) : Icc 0 N = ∅ := byN:ℤh:¬N ≥ 0⊢ Icc 0 N = ∅
  exthN:ℤh:¬N ≥ 0a✝:ℤ⊢ a✝ ∈ Icc 0 N ↔ a✝ ∈ ∅; simphN:ℤh:¬N ≥ 0a✝:ℤ⊢ 0 ≤ a✝ → N < a✝; introshN:ℤh:¬N ≥ 0a✝¹:ℤa✝:0 ≤ a✝¹⊢ N < a✝¹; contrapose! hhN:ℤa✝¹:ℤa✝:0 ≤ a✝¹h:a✝¹ ≤ N⊢ N ≥ 0; linarithAll goals completed! 🐙

Theorem 8.2.2, preliminary version. The arguments here are rearranged slightly from the text.

theorem declaration uses `sorry`sum_of_sum_of_AbsConvergent_nonneg {f:ℕ × ℕ → ℝ} (hf:AbsConvergent f) (hpos: ∀ n m, 0 ≤ f (n, m)) :
  (∀ n, ((fun m ↦ f (n, m)):Series).converges) ∧
  (fun n ↦ ((fun m ↦ f (n, m)):Series).sum:Series).convergesTo (Sum f) := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
  set L := Sum ff:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum f⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    L
  set a : ℕ → Series := fun n ↦ ((fun m ↦ f (n, m)):Series)f:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    L
  have hLpos : 0 ≤ L := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
    simp [L, Sum, hf]f:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }⊢ 0 ≤ { m := 0, seq := fun n ↦ if 0 ≤ n then f (⋯.choose n.toNat) else 0, vanish := ⋯ }.sum; apply sum_of_nonneghnonf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }⊢ { m := 0, seq := fun n ↦ if 0 ≤ n then f (⋯.choose n.toNat) else 0, vanish := ⋯ }.nonneg; intro nhnonf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }n:ℤ⊢ { m := 0, seq := fun n ↦ if 0 ≤ n then f (⋯.choose n.toNat) else 0, vanish := ⋯ }.seq n ≥ 0; by_cases h: n ≥ 0posf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }n:ℤh:n ≥ 0⊢ { m := 0, seq := fun n ↦ if 0 ≤ n then f (⋯.choose n.toNat) else 0, vanish := ⋯ }.seq n ≥ 0negf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }n:ℤh:¬n ≥ 0⊢ { m := 0, seq := fun n ↦ if 0 ≤ n then f (⋯.choose n.toNat) else 0, vanish := ⋯ }.seq n ≥ 0 <;>posf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }n:ℤh:n ≥ 0⊢ { m := 0, seq := fun n ↦ if 0 ≤ n then f (⋯.choose n.toNat) else 0, vanish := ⋯ }.seq n ≥ 0negf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }n:ℤh:¬n ≥ 0⊢ { m := 0, seq := fun n ↦ if 0 ≤ n then f (⋯.choose n.toNat) else 0, vanish := ⋯ }.seq n ≥ 0 simp [h]All goals completed! 🐙; grindAll goals completed! 🐙
  have hfinsum (X: Finset (ℕ × ℕ)) : ∑ p ∈ X, f p ≤ L := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f) sorryAll goals completed! 🐙
  have hfinsum' (n M:ℕ) : (a n).partial M ≤ L := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
    simp [a, Series.partial, Finset.Icc_eq_cast]f:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Ln:ℕM:ℕ⊢ ∑ x ∈ Icc 0 M, f (n, x) ≤ L
    convert_to ∑ x ∈ .map (Embedding.sectR n ℕ) (.Icc 0 M), f x ≤ Lh.e'_3f:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Ln:ℕM:ℕ⊢ ∑ x ∈ Icc 0 M, f (n, x) = ∑ x ∈ Finset.map (Embedding.sectR n ℕ) (Icc 0 M), f xf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Ln:ℕM:ℕ⊢ ∑ x ∈ Finset.map (Embedding.sectR n ℕ) (Icc 0 M), f x ≤ L
    .h.e'_3f:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Ln:ℕM:ℕ⊢ ∑ x ∈ Icc 0 M, f (n, x) = ∑ x ∈ Finset.map (Embedding.sectR n ℕ) (Icc 0 M), f x simpAll goals completed! 🐙
    solve_by_elimAll goals completed! 🐙
  have hnon (n:ℕ) : (a n).nonneg := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
    simp [a, nonneg]f:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Ln:ℕ⊢ ∀ (n_1 : ℤ), 0 ≤ if 0 ≤ n_1 then f (n, n_1.toNat) else 0; intro mf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Ln:ℕm:ℤ⊢ 0 ≤ if 0 ≤ m then f (n, m.toNat) else 0; split_ifsposf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Ln:ℕm:ℤh✝:0 ≤ m⊢ 0 ≤ f (n, m.toNat)negf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Ln:ℕm:ℤh✝:¬0 ≤ m⊢ 0 ≤ 0 <;>posf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Ln:ℕm:ℤh✝:0 ≤ m⊢ 0 ≤ f (n, m.toNat)negf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Ln:ℕm:ℤh✝:¬0 ≤ m⊢ 0 ≤ 0 simp [hpos]All goals completed! 🐙
  have hconv (n:ℕ) : (a n).converges := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
    rw [converges_of_nonneg_iff (hnon n)]f:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonnegn:ℕ⊢ ∃ M, ∀ (N : ℤ), (a n).partial N ≤ M
    use Lhf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonnegn:ℕ⊢ ∀ (N : ℤ), (a n).partial N ≤ L; intro Nhf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonnegn:ℕN:ℤ⊢ (a n).partial N ≤ L; by_cases h: N ≥ 0posf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonnegn:ℕN:ℤh:N ≥ 0⊢ (a n).partial N ≤ Lnegf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonnegn:ℕN:ℤh:¬N ≥ 0⊢ (a n).partial N ≤ L
    .posf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonnegn:ℕN:ℤh:N ≥ 0⊢ (a n).partial N ≤ L lift N to ℕ using hposf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonnegn:ℕN:ℕ⊢ (a n).partial ↑N ≤ L; solve_by_elimAll goals completed! 🐙
    rw [partial_of_lt (by simp [a]; linarith)]negf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonnegn:ℕN:ℤh:¬N ≥ 0⊢ 0 ≤ L; simp [hLpos]All goals completed! 🐙
  have (N M:ℤ) : ∑ n ∈ Icc 0 N, (a n.toNat).partial M ≤ L := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
    by_cases hN : N ≥ 0posf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesN:ℤM:ℤhN:N ≥ 0⊢ ∑ n ∈ Icc 0 N, (a n.toNat).partial M ≤ Lnegf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesN:ℤM:ℤhN:¬N ≥ 0⊢ ∑ n ∈ Icc 0 N, (a n.toNat).partial M ≤ L; swapnegf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesN:ℤM:ℤhN:¬N ≥ 0⊢ ∑ n ∈ Icc 0 N, (a n.toNat).partial M ≤ Lposf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesN:ℤM:ℤhN:N ≥ 0⊢ ∑ n ∈ Icc 0 N, (a n.toNat).partial M ≤ L
    .negf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesN:ℤM:ℤhN:¬N ≥ 0⊢ ∑ n ∈ Icc 0 N, (a n.toNat).partial M ≤ L simp [Finset.Icc_empty hN, hLpos]All goals completed! 🐙
    lift N to ℕ using hNposf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesM:ℤN:ℕ⊢ ∑ n ∈ Icc 0 ↑N, (a n.toNat).partial M ≤ L
    by_cases hM : M ≥ 0posf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesM:ℤN:ℕhM:M ≥ 0⊢ ∑ n ∈ Icc 0 ↑N, (a n.toNat).partial M ≤ Lnegf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesM:ℤN:ℕhM:¬M ≥ 0⊢ ∑ n ∈ Icc 0 ↑N, (a n.toNat).partial M ≤ L; swapnegf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesM:ℤN:ℕhM:¬M ≥ 0⊢ ∑ n ∈ Icc 0 ↑N, (a n.toNat).partial M ≤ Lposf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesM:ℤN:ℕhM:M ≥ 0⊢ ∑ n ∈ Icc 0 ↑N, (a n.toNat).partial M ≤ L
    .negf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesM:ℤN:ℕhM:¬M ≥ 0⊢ ∑ n ∈ Icc 0 ↑N, (a n.toNat).partial M ≤ L convert hLposh.e'_3f:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesM:ℤN:ℕhM:¬M ≥ 0⊢ ∑ n ∈ Icc 0 ↑N, (a n.toNat).partial M = 0; unfold Series.partialh.e'_3f:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesM:ℤN:ℕhM:¬M ≥ 0⊢ ∑ n ∈ Icc 0 ↑N, ∑ n_1 ∈ Icc (a n.toNat).m M, (a n.toNat).seq n_1 = 0
      apply sum_eq_zeroh.e'_3.hf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesM:ℤN:ℕhM:¬M ≥ 0⊢ ∀ x ∈ Icc 0 ↑N, ∑ n ∈ Icc (a x.toNat).m M, (a x.toNat).seq n = 0; intro n _h.e'_3.hf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesM:ℤN:ℕhM:¬M ≥ 0n:ℤa✝:n ∈ Icc 0 ↑N⊢ ∑ n_1 ∈ Icc (a n.toNat).m M, (a n.toNat).seq n_1 = 0
      simp [a, Finset.Icc_empty hM]All goals completed! 🐙
    lift M to ℕ using hMposf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesN:ℕM:ℕ⊢ ∑ n ∈ Icc 0 ↑N, (a n.toNat).partial ↑M ≤ L
    convert_to ∑ x ∈ (Icc 0 N) ×ˢ (.Icc 0 M), f x ≤ Lh.e'_3f:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesN:ℕM:ℕ⊢ ∑ n ∈ Icc 0 ↑N, (a n.toNat).partial ↑M = ∑ x ∈ Icc 0 N ×ˢ Icc 0 M, f xposf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesN:ℕM:ℕ⊢ ∑ x ∈ Icc 0 N ×ˢ Icc 0 M, f x ≤ L
    .h.e'_3f:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesN:ℕM:ℕ⊢ ∑ n ∈ Icc 0 ↑N, (a n.toNat).partial ↑M = ∑ x ∈ Icc 0 N ×ˢ Icc 0 M, f x simp [a, sum_product, Series.partial]All goals completed! 🐙
    solve_by_elimAll goals completed! 🐙
  replace (N:ℤ) : ∑ n ∈ Icc 0 N, (a n.toNat).sum ≤ L := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
    apply le_of_tendsto' (x := .atTop) (tendsto_finset_sum _ _) (this N)f:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesthis:∀ (N M : ℤ), ∑ n ∈ Icc 0 N, (a n.toNat).partial M ≤ LN:ℤ⊢ ∀ i ∈ Icc 0 N, Tendsto (a i.toNat).partial atTop (nhds (a i.toNat).sum)
    solve_by_elim [convergesTo_sum]All goals completed! 🐙
  replace (N:ℤ) : (fun n ↦ (a n).sum:Series).partial N ≤ L := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
    convert this N with n hnh.e'_3.af:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesthis:∀ (N : ℤ), ∑ n ∈ Icc 0 N, (a n.toNat).sum ≤ LN:ℤn:ℤhn:n ∈ Icc 0 N⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.seq n = (a n.toNat).sum; simp_allAll goals completed! 🐙
  have hnon' : (fun n ↦ (a n).sum:Series).nonneg := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
    intro nf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesthis:∀ (N : ℤ), { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.partial N ≤ Ln:ℤ⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.seq n ≥ 0; simpf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesthis:∀ (N : ℤ), { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.partial N ≤ Ln:ℤ⊢ 0 ≤ if 0 ≤ n then (a n.toNat).sum else 0; split_ifsposf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesthis:∀ (N : ℤ), { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.partial N ≤ Ln:ℤh✝:0 ≤ n⊢ 0 ≤ (a n.toNat).sumnegf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesthis:∀ (N : ℤ), { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.partial N ≤ Ln:ℤh✝:¬0 ≤ n⊢ 0 ≤ 0
    .posf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesthis:∀ (N : ℤ), { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.partial N ≤ Ln:ℤh✝:0 ≤ n⊢ 0 ≤ (a n.toNat).sum exact sum_of_nonneg (hnon n.toNat)All goals completed! 🐙
    simpAll goals completed! 🐙
  have hconv' : (fun n ↦ (a n).sum:Series).converges := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
    rw [converges_of_nonneg_iff hnon']f:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergesthis:∀ (N : ℤ), { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.partial N ≤ Lhnon':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.nonneg⊢ ∃ M, ∀ (N : ℤ), { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.partial N ≤ M; use LAll goals completed! 🐙
  replace : (fun n ↦ (a n).sum:Series).sum ≤ L := le_of_tendsto' (convergesTo_sum hconv') thisf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergeshnon':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.nonneghconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.convergesthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.sum ≤ L⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    L
  replace : (fun n ↦ (a n).sum:Series).sum = L := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
    apply le_antisymm this (le_of_forall_sub_le _)f:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergeshnon':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.nonneghconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.convergesthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.sum ≤ L⊢ ∀ ε > 0, L - ε ≤ { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.sum; intro ε hεf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergeshnon':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.nonneghconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.convergesthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.sum ≤ Lε:ℝhε:ε > 0⊢ L - ε ≤ { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.sum
    replace : ∃ X, ∑ p ∈ X, f p ≥ L - ε := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
      sorryAll goals completed! 🐙
    choose X hX using thisf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergeshnon':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.nonneghconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.convergesε:ℝhε:ε > 0X:Finset (ℕ × ℕ)hX:∑ p ∈ X, f p ≥ L - ε⊢ L - ε ≤ { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.sum
    have : ∃ N, ∃ M, X ⊆ (Icc 0 N) ×ˢ (Icc 0 M) := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)⊢ (∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.converges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
      sorryAll goals completed! 🐙
    choose N M hX' using thisf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergeshnon':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.nonneghconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.convergesε:ℝhε:ε > 0X:Finset (ℕ × ℕ)hX:∑ p ∈ X, f p ≥ L - εN:ℕM:ℕhX':X ⊆ Icc 0 N ×ˢ Icc 0 M⊢ L - ε ≤ { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.sum
    calc
      _ ≤ ∑ p ∈ X, f p := hX
      _ ≤ ∑ p ∈ (Icc 0 N) ×ˢ (Icc 0 M), f p := sum_le_sum_of_subset_of_nonneg hX' (byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergeshnon':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.nonneghconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.convergesε:ℝhε:ε > 0X:Finset (ℕ × ℕ)hX:∑ p ∈ X, f p ≥ L - εN:ℕM:ℕhX':X ⊆ Icc 0 N ×ˢ Icc 0 M⊢ ∀ i ∈ Icc 0 N ×ˢ Icc 0 M, i ∉ X → 0 ≤ f i solve_by_elimAll goals completed! 🐙)
      _ = ∑ n ∈ Icc 0 N, ∑ m ∈ Icc 0 M, f (n, m) := sum_product _ _ _
      _ ≤ ∑ n ∈ Icc 0 N, (a n).sum := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergeshnon':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.nonneghconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.convergesε:ℝhε:ε > 0X:Finset (ℕ × ℕ)hX:∑ p ∈ X, f p ≥ L - εN:ℕM:ℕhX':X ⊆ Icc 0 N ×ˢ Icc 0 M⊢ ∑ n ∈ Icc 0 N, ∑ m ∈ Icc 0 M, f (n, m) ≤ ∑ n ∈ Icc 0 N, (a n).sum
        apply sum_le_sumhf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergeshnon':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.nonneghconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.convergesε:ℝhε:ε > 0X:Finset (ℕ × ℕ)hX:∑ p ∈ X, f p ≥ L - εN:ℕM:ℕhX':X ⊆ Icc 0 N ×ˢ Icc 0 M⊢ ∀ i ∈ Icc 0 N, ∑ m ∈ Icc 0 M, f (i, m) ≤ (a i).sum; intro n _hf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergeshnon':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.nonneghconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.convergesε:ℝhε:ε > 0X:Finset (ℕ × ℕ)hX:∑ p ∈ X, f p ≥ L - εN:ℕM:ℕhX':X ⊆ Icc 0 N ×ˢ Icc 0 Mn:ℕa✝:n ∈ Icc 0 N⊢ ∑ m ∈ Icc 0 M, f (n, m) ≤ (a n).sum
        convert partial_le_sum_of_nonneg (hnon n) (hconv n) Mh.e'_3f:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergeshnon':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.nonneghconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.convergesε:ℝhε:ε > 0X:Finset (ℕ × ℕ)hX:∑ p ∈ X, f p ≥ L - εN:ℕM:ℕhX':X ⊆ Icc 0 N ×ˢ Icc 0 Mn:ℕa✝:n ∈ Icc 0 N⊢ ∑ m ∈ Icc 0 M, f (n, m) = (a n).partial ↑M
        simp [a, Series.partial]All goals completed! 🐙
      _ = (fun n ↦ (a n).sum:Series).partial N := byf:ℕ × ℕ → ℝhf:AbsConvergent fhpos:∀ (n m : ℕ), 0 ≤ f (n, m)L:ℝ := Sum fa:ℕ → Series := fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }hLpos:0 ≤ Lhfinsum:∀ (X : Finset (ℕ × ℕ)), ∑ p ∈ X, f p ≤ Lhfinsum':∀ (n M : ℕ), (a n).partial ↑M ≤ Lhnon:∀ (n : ℕ), (a n).nonneghconv:∀ (n : ℕ), (a n).convergeshnon':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.nonneghconv':{ m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.convergesε:ℝhε:ε > 0X:Finset (ℕ × ℕ)hX:∑ p ∈ X, f p ≥ L - εN:ℕM:ℕhX':X ⊆ Icc 0 N ×ˢ Icc 0 M⊢ ∑ n ∈ Icc 0 N, (a n).sum =
  { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ (a n).sum) n.toNat else 0, vanish := ⋯ }.partial ↑N simp [Series.partial]All goals completed! 🐙
      _ ≤ _ := partial_le_sum_of_nonneg hnon' hconv' _
  simp [a, hconv, ← this, Series.convergesTo_sum hconv']All goals completed! 🐙

Theorem 8.2.2, second version

theorem declaration uses `sorry`sum_of_sum_of_AbsConvergent {f:ℕ × ℕ → ℝ} (hf:AbsConvergent f) :
  (∀ n, ((fun m ↦ f (n, m)):Series).absConverges) ∧
  (fun n ↦ ((fun m ↦ f (n, m)):Series).sum:Series).convergesTo (Sum f) := byf:ℕ × ℕ → ℝhf:AbsConvergent f⊢ (∀ (n : ℕ),
    { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.absConverges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
  set fplus := max f 0f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0⊢ (∀ (n : ℕ),
    { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.absConverges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
  set fminus := max (-f) 0f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0⊢ (∀ (n : ℕ),
    { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.absConverges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
  have hfplus_nonneg : ∀ n m, 0 ≤ fplus (n, m) := byf:ℕ × ℕ → ℝhf:AbsConvergent f⊢ (∀ (n : ℕ),
    { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.absConverges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f) intro n mf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0n:ℕm:ℕ⊢ 0 ≤ fplus (n, m); simp [fplus]All goals completed! 🐙
  have hfminus_nonneg : ∀ n m, 0 ≤ fminus (n, m) := byf:ℕ × ℕ → ℝhf:AbsConvergent f⊢ (∀ (n : ℕ),
    { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.absConverges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f) intro n mf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)n:ℕm:ℕ⊢ 0 ≤ fminus (n, m); simp [fminus]All goals completed! 🐙
  have hdiff : f = fplus - fminus := byf:ℕ × ℕ → ℝhf:AbsConvergent f⊢ (∀ (n : ℕ),
    { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.absConverges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f) sorryAll goals completed! 🐙
  have hfplus_conv : AbsConvergent fplus := byf:ℕ × ℕ → ℝhf:AbsConvergent f⊢ (∀ (n : ℕ),
    { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.absConverges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f) sorryAll goals completed! 🐙
  have hfminus_conv : AbsConvergent fminus := byf:ℕ × ℕ → ℝhf:AbsConvergent f⊢ (∀ (n : ℕ),
    { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.absConverges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f) sorryAll goals completed! 🐙
  choose hfplus_conv' hfplus_sum using sum_of_sum_of_AbsConvergent_nonneg hfplus_conv hfplus_nonnegf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)⊢ (∀ (n : ℕ),
    { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.absConverges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
  choose hfminus_conv' hfminus_sum using sum_of_sum_of_AbsConvergent_nonneg hfminus_conv hfminus_nonnegf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)⊢ (∀ (n : ℕ),
    { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.absConverges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
  split_andsrefine_1f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)⊢ ∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.absConvergesrefine_2f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)⊢ { m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum f)
  .refine_1f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)⊢ ∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.absConverges intro nrefine_1f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)n:ℕ⊢ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.absConverges
    sorryAll goals completed! 🐙
  convert convergesTo.sub hfplus_sum hfminus_sum using 1h.e'_1f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)⊢ { m := 0,
    seq := fun n ↦
      if n ≥ 0 then
        (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
          n.toNat
      else 0,
    vanish := ⋯ } =
  { m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ } -
    { m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }h.e'_2f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)⊢ Sum f = Sum fplus - Sum fminus
  .h.e'_1f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)⊢ { m := 0,
    seq := fun n ↦
      if n ≥ 0 then
        (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
          n.toNat
      else 0,
    vanish := ⋯ } =
  { m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ } -
    { m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ } -- encountered surprising difficulty with definitional equivalence here
    simp [hdiff]h.e'_1f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)⊢ { m := 0,
    seq := fun n ↦
      if 0 ≤ n then
        { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fplus (n.toNat, n_1.toNat) - fminus (n.toNat, n_1.toNat) else 0,
            vanish := ⋯ }.sum
      else 0,
    vanish := ⋯ } =
  { m := 0,
      seq := fun n ↦
        if 0 ≤ n then { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fplus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }.sum
        else 0,
      vanish := ⋯ } -
    { m := 0,
      seq := fun n ↦
        if 0 ≤ n then { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fminus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }.sum
        else 0,
      vanish := ⋯ }
    change (fun n ↦ ((fun m ↦ (fplus - fminus) (n, m)):Series).sum:Series) =
      (fun n ↦ ((fun m ↦ fplus (n, m)):Series).sum:Series)
      - (fun n ↦ ((fun m ↦ (fminus) (n, m)):Series).sum:Series)h.e'_1f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)⊢ { m := 0,
    seq := fun n ↦
      if n ≥ 0 then
        (fun n ↦
            { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ (fplus - fminus) (n, m)) n_1.toNat else 0,
                vanish := ⋯ }.sum)
          n.toNat
      else 0,
    vanish := ⋯ } =
  { m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ } -
    { m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }
    convert_to (fun n ↦ ((fun m ↦ (fplus - fminus) (n, m)):Series).sum:Series) =
      (((fun n ↦ ((fun m ↦ fplus (n, m)):Series).sum) - (fun n ↦ ((fun m ↦ (fminus) (n, m)):Series).sum):ℕ → ℝ):Series)h.e'_3f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)⊢ { m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ } -
    { m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ } =
  { m := 0,
    seq := fun n ↦
      if n ≥ 0 then
        ((fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum) -
            fun n ↦
            { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
          n.toNat
      else 0,
    vanish := ⋯ }h.e'_1f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)⊢ { m := 0,
    seq := fun n ↦
      if n ≥ 0 then
        (fun n ↦
            { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ (fplus - fminus) (n, m)) n_1.toNat else 0,
                vanish := ⋯ }.sum)
          n.toNat
      else 0,
    vanish := ⋯ } =
  { m := 0,
    seq := fun n ↦
      if n ≥ 0 then
        ((fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum) -
            fun n ↦
            { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
          n.toNat
      else 0,
    vanish := ⋯ }
    .h.e'_3f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)⊢ { m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ } -
    { m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ } =
  { m := 0,
    seq := fun n ↦
      if n ≥ 0 then
        ((fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum) -
            fun n ↦
            { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
          n.toNat
      else 0,
    vanish := ⋯ } convert sub_coe _ _All goals completed! 🐙
    rcongr _ nh.e'_1.seq.h.e_self.e_seq.h.e_tf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)x✝:ℤn:ℤ⊢ (fun n ↦
      { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ (fplus - fminus) (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
    n.toNat =
  ((fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum) -
      fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
    n.toNat; simph.e'_1.seq.h.e_self.e_seq.h.e_tf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)x✝:ℤn:ℤ⊢ { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fplus (n.toNat, n_1.toNat) - fminus (n.toNat, n_1.toNat) else 0,
      vanish := ⋯ }.sum =
  { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fplus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }.sum -
    { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fminus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }.sum
    convert (sub _ _).2 with mh.e'_2.h.e'_1.h.e'_1f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)x✝:ℤn:ℤ⊢ 0 =
  ({ m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fplus (n.toNat, n_1.toNat) else 0, vanish := ⋯ } -
      { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fminus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }).mh.e'_2.h.e'_1.h.e'_2.hf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)x✝:ℤn:ℤm:ℤ⊢ (if 0 ≤ m then fplus (n.toNat, m.toNat) - fminus (n.toNat, m.toNat) else 0) =
  ({ m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fplus (n.toNat, n_1.toNat) else 0, vanish := ⋯ } -
        { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fminus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }).seq
    mh.e'_1.seq.h.e_self.e_seq.h.e_t.convert_3f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)x✝:ℤn:ℤ⊢ { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fplus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }.convergesh.e'_1.seq.h.e_self.e_seq.h.e_t.convert_4f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)x✝:ℤn:ℤ⊢ { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fminus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }.converges; rflh.e'_2.h.e'_1.h.e'_2.hf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)x✝:ℤn:ℤm:ℤ⊢ (if 0 ≤ m then fplus (n.toNat, m.toNat) - fminus (n.toNat, m.toNat) else 0) =
  ({ m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fplus (n.toNat, n_1.toNat) else 0, vanish := ⋯ } -
        { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fminus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }).seq
    mh.e'_1.seq.h.e_self.e_seq.h.e_t.convert_3f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)x✝:ℤn:ℤ⊢ { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fplus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }.convergesh.e'_1.seq.h.e_self.e_seq.h.e_t.convert_4f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)x✝:ℤn:ℤ⊢ { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fminus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }.converges
    split_ifs with hposf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)x✝:ℤn:ℤm:ℤh:0 ≤ m⊢ fplus (n.toNat, m.toNat) - fminus (n.toNat, m.toNat) =
  ({ m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fplus (n.toNat, n_1.toNat) else 0, vanish := ⋯ } -
        { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fminus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }).seq
    mnegf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)x✝:ℤn:ℤm:ℤh:¬0 ≤ m⊢ 0 =
  ({ m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fplus (n.toNat, n_1.toNat) else 0, vanish := ⋯ } -
        { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fminus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }).seq
    m <;>posf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)x✝:ℤn:ℤm:ℤh:0 ≤ m⊢ fplus (n.toNat, m.toNat) - fminus (n.toNat, m.toNat) =
  ({ m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fplus (n.toNat, n_1.toNat) else 0, vanish := ⋯ } -
        { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fminus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }).seq
    mnegf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)x✝:ℤn:ℤm:ℤh:¬0 ≤ m⊢ 0 =
  ({ m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fplus (n.toNat, n_1.toNat) else 0, vanish := ⋯ } -
        { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fminus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }).seq
    m simp [h, HSub.hSub, Sub.sub]All goals completed! 🐙
    .h.e'_1.seq.h.e_self.e_seq.h.e_t.convert_3f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)x✝:ℤn:ℤ⊢ { m := 0, seq := fun n_1 ↦ if 0 ≤ n_1 then fplus (n.toNat, n_1.toNat) else 0, vanish := ⋯ }.converges solve_by_elimAll goals completed! 🐙
    convert hfminus_conv' n.toNatAll goals completed! 🐙
  have ⟨ g, hg, _ ⟩ := hfh.e'_2f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)g:ℕ → ℕ × ℕhg:Bijective gright✝:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges⊢ Sum f = Sum fplus - Sum fminus
  have h1 := Sum.eq hg (hf.comp hg)h.e'_2f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)g:ℕ → ℕ × ℕhg:Bijective gright✝:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesh1:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum f)⊢ Sum f = Sum fplus - Sum fminus
  have hplus := Sum.eq hg (hfplus_conv.comp hg)h.e'_2f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)g:ℕ → ℕ × ℕhg:Bijective gright✝:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesh1:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum f)hplus:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fplus ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fplus)⊢ Sum f = Sum fplus - Sum fminus
  have hminus := Sum.eq hg (hfminus_conv.comp hg)h.e'_2f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)g:ℕ → ℕ × ℕhg:Bijective gright✝:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesh1:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum f)hplus:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fplus ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fplus)hminus:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fminus ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fminus)⊢ Sum f = Sum fplus - Sum fminus
  apply convergesTo_uniq h1 _f:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)g:ℕ → ℕ × ℕhg:Bijective gright✝:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesh1:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum f)hplus:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fplus ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fplus)hminus:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fminus ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fminus)⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fplus - Sum fminus)
  convert (convergesTo.sub hplus hminus) using 3 with nh.e'_1.h.e'_2.hf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)g:ℕ → ℕ × ℕhg:Bijective gright✝:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesh1:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum f)hplus:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fplus ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fplus)hminus:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fminus ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fminus)n:ℤ⊢ (if n ≥ 0 then (f ∘ g) n.toNat else 0) =
  ({ m := 0, seq := fun n ↦ if n ≥ 0 then (fplus ∘ g) n.toNat else 0, vanish := ⋯ } -
        { m := 0, seq := fun n ↦ if n ≥ 0 then (fminus ∘ g) n.toNat else 0, vanish := ⋯ }).seq
    n
  split_ifs with hposf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)g:ℕ → ℕ × ℕhg:Bijective gright✝:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesh1:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum f)hplus:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fplus ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fplus)hminus:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fminus ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fminus)n:ℤh:n ≥ 0⊢ (f ∘ g) n.toNat =
  ({ m := 0, seq := fun n ↦ if n ≥ 0 then (fplus ∘ g) n.toNat else 0, vanish := ⋯ } -
        { m := 0, seq := fun n ↦ if n ≥ 0 then (fminus ∘ g) n.toNat else 0, vanish := ⋯ }).seq
    nnegf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)g:ℕ → ℕ × ℕhg:Bijective gright✝:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesh1:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum f)hplus:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fplus ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fplus)hminus:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fminus ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fminus)n:ℤh:¬n ≥ 0⊢ 0 =
  ({ m := 0, seq := fun n ↦ if n ≥ 0 then (fplus ∘ g) n.toNat else 0, vanish := ⋯ } -
        { m := 0, seq := fun n ↦ if n ≥ 0 then (fminus ∘ g) n.toNat else 0, vanish := ⋯ }).seq
    n <;>posf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)g:ℕ → ℕ × ℕhg:Bijective gright✝:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesh1:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum f)hplus:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fplus ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fplus)hminus:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fminus ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fminus)n:ℤh:n ≥ 0⊢ (f ∘ g) n.toNat =
  ({ m := 0, seq := fun n ↦ if n ≥ 0 then (fplus ∘ g) n.toNat else 0, vanish := ⋯ } -
        { m := 0, seq := fun n ↦ if n ≥ 0 then (fminus ∘ g) n.toNat else 0, vanish := ⋯ }).seq
    nnegf:ℕ × ℕ → ℝhf:AbsConvergent ffplus:ℕ × ℕ → ℝ := f ⊔ 0fminus:ℕ × ℕ → ℝ := -f ⊔ 0hfplus_nonneg:∀ (n m : ℕ), 0 ≤ fplus (n, m)hfminus_nonneg:∀ (n m : ℕ), 0 ≤ fminus (n, m)hdiff:f = fplus - fminushfplus_conv:AbsConvergent fplushfminus_conv:AbsConvergent fminushfplus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfplus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fplus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fplus)hfminus_conv':∀ (n : ℕ), { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.convergeshfminus_sum:{ m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦
              { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ fminus (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.convergesTo
  (Sum fminus)g:ℕ → ℕ × ℕhg:Bijective gright✝:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConvergesh1:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum f)hplus:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fplus ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fplus)hminus:{ m := 0, seq := fun n ↦ if n ≥ 0 then (fminus ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fminus)n:ℤh:¬n ≥ 0⊢ 0 =
  ({ m := 0, seq := fun n ↦ if n ≥ 0 then (fplus ∘ g) n.toNat else 0, vanish := ⋯ } -
        { m := 0, seq := fun n ↦ if n ≥ 0 then (fminus ∘ g) n.toNat else 0, vanish := ⋯ }).seq
    n simp [h, hdiff, HSub.hSub, Sub.sub]All goals completed! 🐙

Theorem 8.2.2, third version

theorem sum_of_sum_of_AbsConvergent' {f:ℕ × ℕ → ℝ} (hf:AbsConvergent f) :
  (∀ m, ((fun n ↦ f (n, m)):Series).absConverges) ∧
  (fun m ↦ ((fun n ↦ f (n, m)):Series).sum:Series).convergesTo (Sum f) := byf:ℕ × ℕ → ℝhf:AbsConvergent f⊢ (∀ (m : ℕ), { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ f (n, m)) n.toNat else 0, vanish := ⋯ }.absConverges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun m ↦ { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ f (n, m)) n.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
  set π: ℕ × ℕ → ℕ × ℕ := fun p ↦ (p.2, p.1)f:ℕ × ℕ → ℝhf:AbsConvergent fπ:ℕ × ℕ → ℕ × ℕ := fun p ↦ (p.2, p.1)⊢ (∀ (m : ℕ), { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ f (n, m)) n.toNat else 0, vanish := ⋯ }.absConverges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun m ↦ { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ f (n, m)) n.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
  have hπ: Bijective π := Involutive.bijective (congrFun rfl)f:ℕ × ℕ → ℝhf:AbsConvergent fπ:ℕ × ℕ → ℕ × ℕ := fun p ↦ (p.2, p.1)hπ:Bijective π⊢ (∀ (m : ℕ), { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ f (n, m)) n.toNat else 0, vanish := ⋯ }.absConverges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun m ↦ { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ f (n, m)) n.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
  have ⟨ g, hg, hconv ⟩ := hff:ℕ × ℕ → ℝhf:AbsConvergent fπ:ℕ × ℕ → ℕ × ℕ := fun p ↦ (p.2, p.1)hπ:Bijective πg:ℕ → ℕ × ℕhg:Bijective ghconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges⊢ (∀ (m : ℕ), { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ f (n, m)) n.toNat else 0, vanish := ⋯ }.absConverges) ∧
  { m := 0,
        seq := fun n ↦
          if n ≥ 0 then
            (fun m ↦ { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ f (n, m)) n.toNat else 0, vanish := ⋯ }.sum)
              n.toNat
          else 0,
        vanish := ⋯ }.convergesTo
    (Sum f)
  convert sum_of_sum_of_AbsConvergent (f := f ∘ π) _ using 2h.e'_2.h.e'_2f:ℕ × ℕ → ℝhf:AbsConvergent fπ:ℕ × ℕ → ℕ × ℕ := fun p ↦ (p.2, p.1)hπ:Bijective πg:ℕ → ℕ × ℕhg:Bijective ghconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges⊢ Sum f = Sum (f ∘ π)f:ℕ × ℕ → ℝhf:AbsConvergent fπ:ℕ × ℕ → ℕ × ℕ := fun p ↦ (p.2, p.1)hπ:Bijective πg:ℕ → ℕ × ℕhg:Bijective ghconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges⊢ AbsConvergent (f ∘ π)
  .h.e'_2.h.e'_2f:ℕ × ℕ → ℝhf:AbsConvergent fπ:ℕ × ℕ → ℕ × ℕ := fun p ↦ (p.2, p.1)hπ:Bijective πg:ℕ → ℕ × ℕhg:Bijective ghconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges⊢ Sum f = Sum (f ∘ π) exact (Sum.of_comp hf hπ).2All goals completed! 🐙
  refine ⟨ _, hπ.comp hg, ?_ ⟩f:ℕ × ℕ → ℝhf:AbsConvergent fπ:ℕ × ℕ → ℕ × ℕ := fun p ↦ (p.2, p.1)hπ:Bijective πg:ℕ → ℕ × ℕhg:Bijective ghconv:{ m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then ((f ∘ π) ∘ π ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges
  convert hconv using 2All goals completed! 🐙

Theorem 8.2.2, fourth version

theorem sum_comm {f:ℕ × ℕ → ℝ} (hf:AbsConvergent f) :
  (fun n ↦ ((fun m ↦ f (n, m)):Series).sum:Series).sum = (fun m ↦ ((fun n ↦ f (n, m)):Series).sum:Series).sum := byf:ℕ × ℕ → ℝhf:AbsConvergent f⊢ { m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun n ↦ { m := 0, seq := fun n_1 ↦ if n_1 ≥ 0 then (fun m ↦ f (n, m)) n_1.toNat else 0, vanish := ⋯ }.sum)
            n.toNat
        else 0,
      vanish := ⋯ }.sum =
  { m := 0,
      seq := fun n ↦
        if n ≥ 0 then
          (fun m ↦ { m := 0, seq := fun n ↦ if n ≥ 0 then (fun n ↦ f (n, m)) n.toNat else 0, vanish := ⋯ }.sum) n.toNat
        else 0,
      vanish := ⋯ }.sum
  simp [sum_of_converges (sum_of_sum_of_AbsConvergent hf).2,
        sum_of_converges (sum_of_sum_of_AbsConvergent' hf).2]All goals completed! 🐙

Lemma 8.2.3 / Exercise 8.2.1

theorem declaration uses `sorry`AbsConvergent.iff {X:Type} (hX:CountablyInfinite X) (f : X → ℝ) :
  AbsConvergent f ↔ BddAbove ( (fun A ↦ ∑ x ∈ A, |f x|) '' .univ ) := byX:TypehX:CountablyInfinite Xf:X → ℝ⊢ AbsConvergent f ↔ BddAbove ((fun A ↦ ∑ x ∈ A, |f x|) '' Set.univ)
    sorryAll goals completed! 🐙

abbrev AbsConvergent' {X:Type} (f: X → ℝ) : Prop := BddAbove ( (fun A ↦ ∑ x ∈ A, |f x|) '' .univ )

theorem AbsConvergent'.of_finite {X:Type} [Finite X] (f:X → ℝ) : AbsConvergent' f := byX:Typeinst✝:Finite Xf:X → ℝ⊢ AbsConvergent' f
  have _ := Fintype.ofFinite XX:Typeinst✝:Finite Xf:X → ℝx✝:Fintype X⊢ AbsConvergent' f
  simp [bddAbove_def]X:Typeinst✝:Finite Xf:X → ℝx✝:Fintype X⊢ ∃ x, ∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ x; use ∑ x, |f x|hX:Typeinst✝:Finite Xf:X → ℝx✝:Fintype X⊢ ∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ ∑ x, |f x|; intro AhX:Typeinst✝:Finite Xf:X → ℝx✝:Fintype XA:Finset X⊢ ∑ x ∈ A, |f x| ≤ ∑ x, |f x|; apply Finset.sum_le_univ_sum_of_nonnegh.wX:Typeinst✝:Finite Xf:X → ℝx✝:Fintype XA:Finset X⊢ ∀ (x : X), 0 ≤ |f x|; simpAll goals completed! 🐙

Not in textbook, but should have been included.

theorem AbsConvergent'.of_countable {X:Type} (hX:CountablyInfinite X) {f:X → ℝ} :
  AbsConvergent' f ↔ AbsConvergent f := byX:TypehX:CountablyInfinite Xf:X → ℝ⊢ AbsConvergent' f ↔ AbsConvergent f
  constructormpX:TypehX:CountablyInfinite Xf:X → ℝ⊢ AbsConvergent' f → AbsConvergent fmprX:TypehX:CountablyInfinite Xf:X → ℝ⊢ AbsConvergent f → AbsConvergent' f
  .mpX:TypehX:CountablyInfinite Xf:X → ℝ⊢ AbsConvergent' f → AbsConvergent f intro hfmpX:TypehX:CountablyInfinite Xf:X → ℝhf:AbsConvergent' f⊢ AbsConvergent f; simp [bddAbove_def] at hfmpX:TypehX:CountablyInfinite Xf:X → ℝhf:∃ x, ∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ x⊢ AbsConvergent f; choose L hL using hfmpX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ L⊢ AbsConvergent f
    have ⟨ g, hg ⟩ := hX.symmmpX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective g⊢ AbsConvergent f; refine ⟨ g, hg, ?_ ⟩mpX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective g⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.absConverges
    unfold absConvergesmpX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective g⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.abs.converges; rw [converges_of_nonneg_iff]mpX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective g⊢ ∃ M, ∀ (N : ℤ), { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.abs.partial N ≤ MmpX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective g⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.abs.nonneg
    .mpX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective g⊢ ∃ M, ∀ (N : ℤ), { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.abs.partial N ≤ M use LhX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective g⊢ ∀ (N : ℤ), { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.abs.partial N ≤ L; intro NhX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gN:ℤ⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.abs.partial N ≤ L; by_cases hN: N ≥ 0posX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gN:ℤhN:N ≥ 0⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.abs.partial N ≤ LnegX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gN:ℤhN:¬N ≥ 0⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.abs.partial N ≤ L
      .posX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gN:ℤhN:N ≥ 0⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.abs.partial N ≤ L lift N to ℕ using hNposX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gN:ℕ⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.abs.partial ↑N ≤ L
        set g':= Embedding.mk g hg.1posX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gN:ℕg':ℕ ↪ X := { toFun := g, inj' := ⋯ }⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.abs.partial ↑N ≤ L
        convert hL (map g' (Icc 0 N))h.e'_3X:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gN:ℕg':ℕ ↪ X := { toFun := g, inj' := ⋯ }⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.abs.partial ↑N =
  ∑ x ∈ Finset.map g' (Icc 0 N), |f x|
        simp [Series.partial]h.e'_3X:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gN:ℕg':ℕ ↪ X := { toFun := g, inj' := ⋯ }⊢ ∑ x ∈ Icc 0 N, |f (g x)| = ∑ x ∈ Icc 0 N, |f (g' x)|; rflAll goals completed! 🐙
      convert hL ∅h.e'_3X:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gN:ℤhN:¬N ≥ 0⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (f ∘ g) n.toNat else 0, vanish := ⋯ }.abs.partial N = ∑ x ∈ ∅, |f x|
      simph.e'_3X:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gN:ℤhN:¬N ≥ 0⊢ { m := 0, seq := fun n ↦ if 0 ≤ n then f (g n.toNat) else 0, vanish := ⋯ }.abs.partial N = 0; apply partial_of_lth.e'_3.hX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gN:ℤhN:¬N ≥ 0⊢ N < { m := 0, seq := fun n ↦ if 0 ≤ n then f (g n.toNat) else 0, vanish := ⋯ }.abs.m; grindAll goals completed! 🐙
    simp [nonneg]mpX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective g⊢ ∀ (n : ℤ), 0 ≤ if 0 ≤ n then |if 0 ≤ n then f (g n.toNat) else 0| else 0
    intro nmpX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gn:ℤ⊢ 0 ≤ if 0 ≤ n then |if 0 ≤ n then f (g n.toNat) else 0| else 0; by_cases h: n ≥ 0posX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gn:ℤh:n ≥ 0⊢ 0 ≤ if 0 ≤ n then |if 0 ≤ n then f (g n.toNat) else 0| else 0negX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gn:ℤh:¬n ≥ 0⊢ 0 ≤ if 0 ≤ n then |if 0 ≤ n then f (g n.toNat) else 0| else 0 <;>posX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gn:ℤh:n ≥ 0⊢ 0 ≤ if 0 ≤ n then |if 0 ≤ n then f (g n.toNat) else 0| else 0negX:TypehX:CountablyInfinite Xf:X → ℝL:ℝhL:∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ Lg:ℕ → Xhg:Bijective gn:ℤh:¬n ≥ 0⊢ 0 ≤ if 0 ≤ n then |if 0 ≤ n then f (g n.toNat) else 0| else 0 simp [h]All goals completed! 🐙
  intro hfmprX:TypehX:CountablyInfinite Xf:X → ℝhf:AbsConvergent f⊢ AbsConvergent' f; rwa [AbsConvergent.iff hX f]mprX:TypehX:CountablyInfinite Xf:X → ℝhf:BddAbove ((fun A ↦ ∑ x ∈ A, |f x|) '' Set.univ)⊢ AbsConvergent' f at hf

Lemma 8.2.5 / Exercise 8.2.2

theorem declaration uses `sorry`AbsConvergent'.countable_supp {X:Type} {f:X → ℝ} (hf: AbsConvergent' f) :
  AtMostCountable { x | f x ≠ 0 } := byX:Typef:X → ℝhf:AbsConvergent' f⊢ AtMostCountable ↑{x | f x ≠ 0}
    sorryAll goals completed! 🐙

Compare with Mathlib's Summable.subtype

theorem AbsConvergent'.subtype {X:Type} {f:X → ℝ} (hf: AbsConvergent' f) (A: Set X) :
  AbsConvergent' (fun x:A ↦ f x) := byX:Typef:X → ℝhf:AbsConvergent' fA:Set X⊢ AbsConvergent' fun x ↦ f ↑x
  apply BddAbove.mono _ hfX:Typef:X → ℝhf:AbsConvergent' fA:Set X⊢ (fun A_1 ↦ ∑ x ∈ A_1, |(fun x ↦ f ↑x) x|) '' Set.univ ⊆ (fun A ↦ ∑ x ∈ A, |f x|) '' Set.univ
  intro z hzX:Typef:X → ℝhf:AbsConvergent' fA:Set Xz:ℝhz:z ∈ (fun A_1 ↦ ∑ x ∈ A_1, |(fun x ↦ f ↑x) x|) '' Set.univ⊢ z ∈ (fun A ↦ ∑ x ∈ A, |f x|) '' Set.univ; simp at *X:Typef:X → ℝhf:AbsConvergent' fA:Set Xz:ℝhz:∃ y, ∑ x ∈ y, |f ↑x| = z⊢ ∃ y, ∑ x ∈ y, |f x| = z; choose A hA using hzX:Typef:X → ℝhf:AbsConvergent' fA✝:Set Xz:ℝA:Finset ↑A✝hA:∑ x ∈ A, |f ↑x| = z⊢ ∃ y, ∑ x ∈ y, |f x| = z
  use A.map (Embedding.subtype _)hX:Typef:X → ℝhf:AbsConvergent' fA✝:Set Xz:ℝA:Finset ↑A✝hA:∑ x ∈ A, |f ↑x| = z⊢ ∑ x ∈ Finset.map (Embedding.subtype fun x ↦ x ∈ A✝) A, |f x| = z; simp [hA]All goals completed! 🐙

A generalized sum. Note that this will give junk values if f is not AbsConvergent'.

noncomputable abbrev Sum' {X:Type} (f: X → ℝ) : ℝ := Sum (fun x : { x | f x ≠ 0 } ↦ f x)

Not in textbook, but should have been included (the series laws are significantly harder to establish without this)

theorem Sum'.of_finsupp {X:Type} {f:X → ℝ} {A: Finset X} (h: ∀ x ∉ A, f x = 0) : Sum' f = ∑ x ∈ A, f x := byX:Typef:X → ℝA:Finset Xh:∀ x ∉ A, f x = 0⊢ Sum' f = ∑ x ∈ A, f x
  unfold Sum'X:Typef:X → ℝA:Finset Xh:∀ x ∉ A, f x = 0⊢ (Sum fun x ↦ f ↑x) = ∑ x ∈ A, f x
  set E := { x | f x ≠ 0 }X:Typef:X → ℝA:Finset Xh:∀ x ∉ A, f x = 0E:Set X := {x | f x ≠ 0}⊢ (Sum fun x ↦ f ↑x) = ∑ x ∈ A, f x
  have hE : E ⊆ A := byX:Typef:X → ℝA:Finset Xh:∀ x ∉ A, f x = 0⊢ Sum' f = ∑ x ∈ A, f x intro _X:Typef:X → ℝA:Finset Xh:∀ x ∉ A, f x = 0E:Set X := {x | f x ≠ 0}a✝:X⊢ a✝ ∈ E → a✝ ∈ ↑A; simp [E]X:Typef:X → ℝA:Finset Xh:∀ x ∉ A, f x = 0E:Set X := {x | f x ≠ 0}a✝:X⊢ ¬f a✝ = 0 → a✝ ∈ A; grindAll goals completed! 🐙
  have hfin : Finite E := Finite.Set.subset _ hEX:Typef:X → ℝA:Finset Xh:∀ x ∉ A, f x = 0E:Set X := {x | f x ≠ 0}hE:E ⊆ ↑Ahfin:Finite ↑E⊢ (Sum fun x ↦ f ↑x) = ∑ x ∈ A, f x
  set E' := E.toFinite.toFinsetX:Typef:X → ℝA:Finset Xh:∀ x ∉ A, f x = 0E:Set X := {x | f x ≠ 0}hE:E ⊆ ↑Ahfin:Finite ↑EE':Finset X := ⋯.toFinset⊢ (Sum fun x ↦ f ↑x) = ∑ x ∈ A, f x
  rw [Sum.of_finite (fun x:E ↦ f x), ←E'.sum_subtype (by simp [E'])]X:Typef:X → ℝA:Finset Xh:∀ x ∉ A, f x = 0E:Set X := {x | f x ≠ 0}hE:E ⊆ ↑Ahfin:Finite ↑EE':Finset X := ⋯.toFinset⊢ ∑ a ∈ E', f a = ∑ x ∈ A, f x
  replace hE : E' ⊆ A := byX:Typef:X → ℝA:Finset Xh:∀ x ∉ A, f x = 0⊢ Sum' f = ∑ x ∈ A, f x aesopAll goals completed! 🐙
  apply sum_subset hEX:Typef:X → ℝA:Finset Xh:∀ x ∉ A, f x = 0E:Set X := {x | f x ≠ 0}hfin:Finite ↑EE':Finset X := ⋯.toFinsethE:E' ⊆ A⊢ ∀ x ∈ A, x ∉ E' → f x = 0; aesopAll goals completed! 🐙

Not in textbook, but should have been included (the series laws are significantly harder to establish without this)

theorem Sum'.of_countable_supp {X:Type} {f:X → ℝ} {A: Set X} (hA: CountablyInfinite A)
  (hfA : ∀ x ∉ A, f x = 0) (hconv: AbsConvergent' f):
  AbsConvergent' (fun x:A ↦ f x) ∧ Sum' f = Sum (fun x:A ↦ f x) := byX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' f⊢ (AbsConvergent' fun x ↦ f ↑x) ∧ Sum' f = Sum fun x ↦ f ↑x
  -- We can adapt the proof of `AbsConvergent'.of_countable` to establish absolute convergence on A.
  have hconv' : AbsConvergent (fun x:A ↦ f x) :=
    (AbsConvergent'.of_countable hA).mp (hconv.subtype A)X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑x⊢ (AbsConvergent' fun x ↦ f ↑x) ∧ Sum' f = Sum fun x ↦ f ↑x
  rw [AbsConvergent'.of_countable hA]X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑x⊢ (AbsConvergent fun x ↦ f ↑x) ∧ Sum' f = Sum fun x ↦ f ↑x
  refine ⟨ hconv', ?_ ⟩X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑x⊢ Sum' f = Sum fun x ↦ f ↑x
  set E := { x | f x ≠ 0 }X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}⊢ Sum' f = Sum fun x ↦ f ↑x
  -- The main challenge here is to relate a sum on E with a sum on A.  First, we show containment.
  have hE : E ⊆ A := byX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' f⊢ (AbsConvergent' fun x ↦ f ↑x) ∧ Sum' f = Sum fun x ↦ f ↑x intro _X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}a✝:X⊢ a✝ ∈ E → a✝ ∈ A; simp [E]X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}a✝:X⊢ ¬f a✝ = 0 → a✝ ∈ A; by_contra!X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}a✝:Xthis:f a✝ ≠ 0 ∧ a✝ ∉ A⊢ False; aesopAll goals completed! 🐙
  -- Now, we map A back to the natural numbers, thus identifying E with a subset E' of ℕ.
  choose g hg using hA.symmX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective g⊢ Sum' f = Sum fun x ↦ f ↑x
  have hsum := Sum.eq hg (hconv'.comp hg)X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)⊢ Sum' f = Sum fun x ↦ f ↑x
  set E' := { n | ↑(g n) ∈ E }X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}⊢ Sum' f = Sum fun x ↦ f ↑x
  set ι : E' → E := fun ⟨ n, hn ⟩ ↦ ⟨ g n, byX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}x✝:↑E'n:ℕhn:n ∈ E'⊢ ↑(g n) ∈ E aesopAll goals completed! 🐙 ⟩
  have hι: Bijective ι := byX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' f⊢ (AbsConvergent' fun x ↦ f ↑x) ∧ Sum' f = Sum fun x ↦ f ↑x
    split_andsrefine_1X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩⊢ Injective ιrefine_2X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩⊢ Surjective ι
    .refine_1X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩⊢ Injective ι intro ⟨ _, _ ⟩ ⟨ _, _ ⟩refine_1X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩val✝¹:ℕproperty✝¹:val✝¹ ∈ E'val✝:ℕproperty✝:val✝ ∈ E'⊢ ι ⟨val✝¹, property✝¹⟩ = ι ⟨val✝, property✝⟩ → ⟨val✝¹, property✝¹⟩ = ⟨val✝, property✝⟩ hrefine_1X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩val✝¹:ℕproperty✝¹:val✝¹ ∈ E'val✝:ℕproperty✝:val✝ ∈ E'h:ι ⟨val✝¹, property✝¹⟩ = ι ⟨val✝, property✝⟩⊢ ⟨val✝¹, property✝¹⟩ = ⟨val✝, property✝⟩; simp [ι, E', Subtype.val_inj] at *refine_1X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective gE':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩val✝¹:ℕproperty✝¹:val✝¹ ∈ E'val✝:ℕproperty✝:val✝ ∈ E'hsum:{ m := 0, seq := fun n ↦ if 0 ≤ n then f ↑(g n.toNat) else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)h:g val✝¹ = g val✝⊢ val✝¹ = val✝; exact hg.1 hAll goals completed! 🐙
    .refine_2X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩⊢ Surjective ι intro ⟨ x, hx ⟩refine_2X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩x:Xhx:x ∈ E⊢ ∃ a, ι a = ⟨x, hx⟩; choose n hn using hg.2 ⟨ _, hE hx ⟩refine_2X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩x:Xhx:x ∈ En:ℕhn:g n = ⟨x, ⋯⟩⊢ ∃ a, ι a = ⟨x, hx⟩; use ⟨ n, byX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩x:Xhx:x ∈ En:ℕhn:g n = ⟨x, ⋯⟩⊢ n ∈ E' aesopAll goals completed! 🐙 ⟩; grindAll goals completed! 🐙
  -- The cases of infinite and finite E' are handled separately.
  obtain hE' | hE' := Nat.atMostCountable_subset E'inlX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'⊢ Sum' f = Sum fun x ↦ f ↑xinrX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'⊢ Sum' f = Sum fun x ↦ f ↑x
  .inlX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'⊢ Sum' f = Sum fun x ↦ f ↑x --   use Nat.monotone_enum_of_infinite to enumerate E'
    --   show the partial sums of E' are a subsequence of the partial sums of A
    set hinf : Infinite E' := hE'.toInfiniteinlX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···⊢ Sum' f = Sum fun x ↦ f ↑x
    choose a ha_bij ha_mono using (Nat.monotone_enum_of_infinite E').existsinlX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono a⊢ Sum' f = Sum fun x ↦ f ↑x
    have : atTop.Tendsto (Nat.cast ∘ Subtype.val ∘ a: ℕ → ℤ) atTop := byX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' f⊢ (AbsConvergent' fun x ↦ f ↑x) ∧ Sum' f = Sum fun x ↦ f ↑x
      apply tendsto_natCast_atTop_atTop.comp (StrictMono.tendsto_atTop _)X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono a⊢ StrictMono (Subtype.val ∘ a)
      intro _ _X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono aa✝:ℕb✝:ℕ⊢ a✝ < b✝ → (Subtype.val ∘ a) a✝ < (Subtype.val ∘ a) b✝ hnmX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono aa✝:ℕb✝:ℕhnm:a✝ < b✝⊢ (Subtype.val ∘ a) a✝ < (Subtype.val ∘ a) b✝; simp [ha_mono hnm]All goals completed! 🐙
    apply tendsto_nhds_unique  _ (hsum.comp this)X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono athis:Tendsto (Nat.cast ∘ Subtype.val ∘ a) atTop atTop⊢ Tendsto
  ({ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.partial ∘
    Nat.cast ∘ Subtype.val ∘ a)
  atTop (nhds (Sum' f))
    have hconv'' : AbsConvergent (fun x:E ↦ f x) := byX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' f⊢ (AbsConvergent' fun x ↦ f ↑x) ∧ Sum' f = Sum fun x ↦ f ↑x
      rw [←AbsConvergent'.of_countable]X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono athis:Tendsto (Nat.cast ∘ Subtype.val ∘ a) atTop atTop⊢ AbsConvergent' fun x ↦ f ↑xhXX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono athis:Tendsto (Nat.cast ∘ Subtype.val ∘ a) atTop atTop⊢ CountablyInfinite ↑E
      .X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono athis:Tendsto (Nat.cast ∘ Subtype.val ∘ a) atTop atTop⊢ AbsConvergent' fun x ↦ f ↑x exact hconv.subtype EAll goals completed! 🐙
      apply (CountablyInfinite.equiv _).mp hE'X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono athis:Tendsto (Nat.cast ∘ Subtype.val ∘ a) atTop atTop⊢ EqualCard ↑E' ↑E; use ιAll goals completed! 🐙
    replace := Sum.eq (hι.comp ha_bij) (hconv''.comp (hι.comp ha_bij))X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)⊢ Tendsto
  ({ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.partial ∘
    Nat.cast ∘ Subtype.val ∘ a)
  atTop (nhds (Sum' f))
    convert this.comp tendsto_natCast_atTop_atTop using 1h.e'_3X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.partial ∘
    Nat.cast ∘ Subtype.val ∘ a =
  { m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.partial ∘ Nat.cast; ext Nh.e'_3.hX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕ⊢ ({ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.partial ∘
      Nat.cast ∘ Subtype.val ∘ a)
    N =
  ({ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.partial ∘ Nat.cast) N
    simp [Series.partial, ι]h.e'_3.hX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕ⊢ ∑ x ∈ Icc 0 ↑(a N), f ↑(g x) = ∑ x ∈ Icc 0 N, f ↑(g ↑(a x))
    calc
      _ = ∑ x ∈ .image (Subtype.val ∘ a) (.Icc 0 N), f ↑(g x) := byX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕ⊢ ∑ x ∈ Icc 0 ↑(a N), f ↑(g x) = ∑ x ∈ image (Subtype.val ∘ a) (Icc 0 N), f ↑(g x)
        symmX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕ⊢ ∑ x ∈ image (Subtype.val ∘ a) (Icc 0 N), f ↑(g x) = ∑ x ∈ Icc 0 ↑(a N), f ↑(g x); apply sum_subsethX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕ⊢ image (Subtype.val ∘ a) (Icc 0 N) ⊆ Icc 0 ↑(a N)hfX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕ⊢ ∀ x ∈ Icc 0 ↑(a N), x ∉ image (Subtype.val ∘ a) (Icc 0 N) → f ↑(g x) = 0
        .hX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕ⊢ image (Subtype.val ∘ a) (Icc 0 N) ⊆ Icc 0 ↑(a N) intro m hmhX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕm:ℕhm:m ∈ image (Subtype.val ∘ a) (Icc 0 N)⊢ m ∈ Icc 0 ↑(a N); simp at hm ⊢hX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕm:ℕhm:∃ a_1 ≤ N, ↑(a a_1) = m⊢ m ≤ ↑(a N); obtain ⟨ n, hn, rfl ⟩ := hmhX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕn:ℕhn:n ≤ N⊢ ↑(a n) ≤ ↑(a N)
          simp [ha_mono.monotone hn]All goals completed! 🐙
        intro x hxhfX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕx:ℕhx:x ∈ Icc 0 ↑(a N)⊢ x ∉ image (Subtype.val ∘ a) (Icc 0 N) → f ↑(g x) = 0 hx'hfX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕx:ℕhx:x ∈ Icc 0 ↑(a N)hx':x ∉ image (Subtype.val ∘ a) (Icc 0 N)⊢ f ↑(g x) = 0; simp at hx hx'hfX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕx:ℕhx:x ≤ ↑(a N)hx':∀ x_1 ≤ N, ¬↑(a x_1) = x⊢ f ↑(g x) = 0; contrapose! hx'hfX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕx:ℕhx:x ≤ ↑(a N)hx':f ↑(g x) ≠ 0⊢ ∃ x_1 ≤ N, ↑(a x_1) = x
        choose n hn using (hι.comp ha_bij).2 ⟨ g x, hx' ⟩hfX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕx:ℕhx:x ≤ ↑(a N)hx':f ↑(g x) ≠ 0n:ℕhn:(ι ∘ a) n = ⟨↑(g x), hx'⟩⊢ ∃ x_1 ≤ N, ↑(a x_1) = x
        simp [ι, Subtype.val_inj] at hnhfX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕx:ℕhx:x ≤ ↑(a N)hx':f ↑(g x) ≠ 0n:ℕhn:g ↑(a n) = g x⊢ ∃ x_1 ≤ N, ↑(a x_1) = x
        apply hg.1 at hnhfX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕx:ℕhx:x ≤ ↑(a N)hx':f ↑(g x) ≠ 0n:ℕhn:↑(a n) = x⊢ ∃ x_1 ≤ N, ↑(a x_1) = x; subst hnhfX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕn:ℕhx:↑(a n) ≤ ↑(a N)hx':f ↑(g ↑(a n)) ≠ 0⊢ ∃ x ≤ N, ↑(a x) = ↑(a n)
        use nhX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕn:ℕhx:↑(a n) ≤ ↑(a N)hx':f ↑(g ↑(a n)) ≠ 0⊢ n ≤ N ∧ ↑(a n) = ↑(a n); simpa [ha_mono.le_iff_le] using hxAll goals completed! 🐙
      _ = _ := byX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕ⊢ ∑ x ∈ image (Subtype.val ∘ a) (Icc 0 N), f ↑(g x) = ∑ x ∈ Icc 0 N, f ↑(g ↑(a x))
        apply sum_imageaX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕ⊢ Set.InjOn (Subtype.val ∘ a) ↑(Icc 0 N)
        intro _ _aX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕx₁✝:ℕa✝:x₁✝ ∈ ↑(Icc 0 N)⊢ ∀ ⦃x₂ : ℕ⦄, x₂ ∈ ↑(Icc 0 N) → (Subtype.val ∘ a) x₁✝ = (Subtype.val ∘ a) x₂ → x₁✝ = x₂ _aX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕx₁✝:ℕa✝:x₁✝ ∈ ↑(Icc 0 N)x₂✝:ℕ⊢ x₂✝ ∈ ↑(Icc 0 N) → (Subtype.val ∘ a) x₁✝ = (Subtype.val ∘ a) x₂✝ → x₁✝ = x₂✝ _aX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕx₁✝:ℕa✝¹:x₁✝ ∈ ↑(Icc 0 N)x₂✝:ℕa✝:x₂✝ ∈ ↑(Icc 0 N)⊢ (Subtype.val ∘ a) x₁✝ = (Subtype.val ∘ a) x₂✝ → x₁✝ = x₂✝ haX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕx₁✝:ℕa✝¹:x₁✝ ∈ ↑(Icc 0 N)x₂✝:ℕa✝:x₂✝ ∈ ↑(Icc 0 N)h:(Subtype.val ∘ a) x₁✝ = (Subtype.val ∘ a) x₂✝⊢ x₁✝ = x₂✝; simp [Subtype.val_inj] at haX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':CountablyInfinite ↑E'hinf:Infinite ↑E' := ···a:ℕ → ↑E'ha_bij:Bijective aha_mono:StrictMono ahconv'':AbsConvergent fun x ↦ f ↑xthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ ι ∘ a) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum fun x ↦ f ↑x)N:ℕx₁✝:ℕa✝¹:x₁✝ ∈ ↑(Icc 0 N)x₂✝:ℕa✝:x₂✝ ∈ ↑(Icc 0 N)h:a x₁✝ = a x₂✝⊢ x₁✝ = x₂✝; exact ha_bij.1 hAll goals completed! 🐙
  -- When E' is finite, we show that all sufficiently large partial sums of A are equal to
  -- the sum of E'.
  let hEfin : Finite E := hι.finite_iff.mp hE'inrX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···⊢ Sum' f = Sum fun x ↦ f ↑x
  let hE'fintype : Fintype E' := .ofFinite _inrX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'⊢ Sum' f = Sum fun x ↦ f ↑x
  let hEfintype : Fintype E := .ofFinite _inrX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑E⊢ Sum' f = Sum fun x ↦ f ↑x
  apply convergesTo_uniq _ hsumX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑E⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum' f)
  simp [Sum.of_finite, Series.convergesTo]X:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑E⊢ Tendsto { m := 0, seq := fun n ↦ if 0 ≤ n then f ↑(g n.toNat) else 0, vanish := ⋯ }.partial atTop (nhds (∑ x, f ↑x))
  apply tendsto_nhds_of_eventually_eqhX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑E⊢ ∀ᶠ (x' : ℤ) in atTop, { m := 0, seq := fun n ↦ if 0 ≤ n then f ↑(g n.toNat) else 0, vanish := ⋯ }.partial x' = ∑ x, f ↑x
  have hE'bound : BddAbove E' := Set.Finite.bddAbove hE'hX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EhE'bound:BddAbove E'⊢ ∀ᶠ (x' : ℤ) in atTop, { m := 0, seq := fun n ↦ if 0 ≤ n then f ↑(g n.toNat) else 0, vanish := ⋯ }.partial x' = ∑ x, f ↑x
  rw [bddAbove_def] at hE'boundhX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EhE'bound:∃ x, ∀ y ∈ E', y ≤ x⊢ ∀ᶠ (x' : ℤ) in atTop, { m := 0, seq := fun n ↦ if 0 ≤ n then f ↑(g n.toNat) else 0, vanish := ⋯ }.partial x' = ∑ x, f ↑x; choose N hN using hE'boundhX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ N⊢ ∀ᶠ (x' : ℤ) in atTop, { m := 0, seq := fun n ↦ if 0 ≤ n then f ↑(g n.toNat) else 0, vanish := ⋯ }.partial x' = ∑ x, f ↑x
  rw [eventually_atTop]hX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ N⊢ ∃ a, ∀ b ≥ a, { m := 0, seq := fun n ↦ if 0 ≤ n then f ↑(g n.toNat) else 0, vanish := ⋯ }.partial b = ∑ x, f ↑x
  use NhX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ N⊢ ∀ b ≥ ↑N, { m := 0, seq := fun n ↦ if 0 ≤ n then f ↑(g n.toNat) else 0, vanish := ⋯ }.partial b = ∑ x, f ↑x; intro N' hN'hX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ NN':ℤhN':N' ≥ ↑N⊢ { m := 0, seq := fun n ↦ if 0 ≤ n then f ↑(g n.toNat) else 0, vanish := ⋯ }.partial N' = ∑ x, f ↑x
  lift N' to ℕ using (LE.le.trans (byX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ NN':ℤhN':N' ≥ ↑N⊢ 0 ≤ ↑N positivityAll goals completed! 🐙) hN')
  simp [Series.partial] at hN' ⊢hX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ NN':ℕhN':N ≤ N'⊢ ∑ x ∈ Icc 0 N', f ↑(g x) = ∑ x, f ↑x
  calc
    _ = ∑ n ∈ E', f (g n) := byX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ NN':ℕhN':N ≤ N'⊢ ∑ x ∈ Icc 0 N', f ↑(g x) = ∑ n ∈ E'.toFinset, f ↑(g n)
      symmX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ NN':ℕhN':N ≤ N'⊢ ∑ n ∈ E'.toFinset, f ↑(g n) = ∑ x ∈ Icc 0 N', f ↑(g x); apply sum_subsethX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ NN':ℕhN':N ≤ N'⊢ E'.toFinset ⊆ Icc 0 N'hfX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ NN':ℕhN':N ≤ N'⊢ ∀ x ∈ Icc 0 N', x ∉ E'.toFinset → f ↑(g x) = 0
      .hX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ NN':ℕhN':N ≤ N'⊢ E'.toFinset ⊆ Icc 0 N' intro x hxhX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ NN':ℕhN':N ≤ N'x:ℕhx:x ∈ E'.toFinset⊢ x ∈ Icc 0 N'; simp at *hX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective gE':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ NN':ℕhN':N ≤ N'x:ℕhsum:{ m := 0, seq := fun n ↦ if 0 ≤ n then f ↑(g n.toNat) else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)hx:x ∈ E'⊢ x ≤ N'; linarith [hN _ hx]All goals completed! 🐙
      intro _ _hfX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ NN':ℕhN':N ≤ N'x✝:ℕa✝:x✝ ∈ Icc 0 N'⊢ x✝ ∉ E'.toFinset → f ↑(g x✝) = 0 hx'hfX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ NN':ℕhN':N ≤ N'x✝:ℕa✝:x✝ ∈ Icc 0 N'hx':x✝ ∉ E'.toFinset⊢ f ↑(g x✝) = 0; simpa [E',E] using hx'All goals completed! 🐙
    _ = ∑ n:E', f (g n) := byX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ NN':ℕhN':N ≤ N'⊢ ∑ n ∈ E'.toFinset, f ↑(g n) = ∑ n, f ↑(g ↑n) convert (sum_set_coe _).symmAll goals completed! 🐙
    _ = ∑ n, f (ι n) := sum_congr rfl (byX:Typef:X → ℝA:Set XhA:CountablyInfinite ↑AhfA:∀ x ∉ A, f x = 0hconv:AbsConvergent' fhconv':AbsConvergent fun x ↦ f ↑xE:Set X := {x | f x ≠ 0}hE:E ⊆ Ag:ℕ → ↑Ahg:Bijective ghsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((fun x ↦ f ↑x) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo (Sum fun x ↦ f ↑x)E':Set ℕ := {n | ↑(g n) ∈ E}ι:↑E' → ↑E := 
  fun x ↦
    match x with
    | ⟨n, hn⟩ => ⟨↑(g n), ⋯⟩hι:Bijective ιhE':Finite ↑E'hEfin:Finite ↑E := ···hE'fintype:Fintype ↑E' := Fintype.ofFinite ↑E'hEfintype:Fintype ↑E := Fintype.ofFinite ↑EN:ℕhN:∀ y ∈ E', y ≤ NN':ℕhN':N ≤ N'⊢ ∀ x ∈ univ, f ↑(g ↑x) = f ↑(ι x) grindAll goals completed! 🐙)
    _ = _ := hι.sum_comp (g := fun x ↦ f x)

Connection with Mathlib's Summable property. Some version of this might be suitable for Mathlib?

theorem AbsConvergent'.iff_Summable {X:Type} (f:X → ℝ) : AbsConvergent' f ↔ Summable f := byX:Typef:X → ℝ⊢ AbsConvergent' f ↔ Summable f
  simp [←summable_abs_iff, AbsConvergent']X:Typef:X → ℝ⊢ BddAbove (Set.range fun A ↦ ∑ x ∈ A, |f x|) ↔ Summable fun x ↦ |(|f x|)|
  simp [summable_iff_vanishing_norm]X:Typef:X → ℝ⊢ BddAbove (Set.range fun A ↦ ∑ x ∈ A, |f x|) ↔
  ∀ (ε : ℝ), 0 < ε → ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < ε
  classical
  constructormpX:Typef:X → ℝ⊢ BddAbove (Set.range fun A ↦ ∑ x ∈ A, |f x|) →
  ∀ (ε : ℝ), 0 < ε → ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < εmprX:Typef:X → ℝ⊢ (∀ (ε : ℝ), 0 < ε → ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < ε) →
  BddAbove (Set.range fun A ↦ ∑ x ∈ A, |f x|)
  .mpX:Typef:X → ℝ⊢ BddAbove (Set.range fun A ↦ ∑ x ∈ A, |f x|) →
  ∀ (ε : ℝ), 0 < ε → ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < ε intro h εmpX:Typef:X → ℝh:BddAbove (Set.range fun A ↦ ∑ x ∈ A, |f x|)ε:ℝ⊢ 0 < ε → ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < ε hεmpX:Typef:X → ℝh:BddAbove (Set.range fun A ↦ ∑ x ∈ A, |f x|)ε:ℝhε:0 < ε⊢ ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < ε
    set s := Set.range fun A ↦ ∑ x ∈ A, |f x|mpX:Typef:X → ℝε:ℝhε:0 < εs:Set ℝ := Set.range fun A ↦ ∑ x ∈ A, |f x|h:BddAbove s⊢ ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < ε
    have hnon : s.Nonempty := byX:Typef:X → ℝ⊢ AbsConvergent' f ↔ Summable f simp [s]X:Typef:X → ℝε:ℝhε:0 < εs:Set ℝ := Set.range fun A ↦ ∑ x ∈ A, |f x|h:BddAbove s⊢ (Set.range fun A ↦ ∑ x ∈ A, |f x|).Nonempty; use 0, ∅hX:Typef:X → ℝε:ℝhε:0 < εs:Set ℝ := Set.range fun A ↦ ∑ x ∈ A, |f x|h:BddAbove s⊢ (fun A ↦ ∑ x ∈ A, |f x|) ∅ = 0; simpAll goals completed! 🐙
    have : (sSup s)-ε < sSup s := byX:Typef:X → ℝ⊢ AbsConvergent' f ↔ Summable f linarithAll goals completed! 🐙
    simp [lt_csSup_iff h hnon,s] at thismpX:Typef:X → ℝε:ℝhε:0 < εs:Set ℝ := Set.range fun A ↦ ∑ x ∈ A, |f x|h:BddAbove shnon:s.Nonemptythis:∃ a, sSup (Set.range fun A ↦ ∑ x ∈ A, |f x|) - ε < ∑ x ∈ a, |f x|⊢ ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < ε; choose S hS using thismpX:Typef:X → ℝε:ℝhε:0 < εs:Set ℝ := Set.range fun A ↦ ∑ x ∈ A, |f x|h:BddAbove shnon:s.NonemptyS:Finset XhS:sSup (Set.range fun A ↦ ∑ x ∈ A, |f x|) - ε < ∑ x ∈ S, |f x|⊢ ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < ε
    use ShX:Typef:X → ℝε:ℝhε:0 < εs:Set ℝ := Set.range fun A ↦ ∑ x ∈ A, |f x|h:BddAbove shnon:s.NonemptyS:Finset XhS:sSup (Set.range fun A ↦ ∑ x ∈ A, |f x|) - ε < ∑ x ∈ S, |f x|⊢ ∀ (t : Finset X), Disjoint t S → |∑ x ∈ t, |f x|| < ε; intro T hThX:Typef:X → ℝε:ℝhε:0 < εs:Set ℝ := Set.range fun A ↦ ∑ x ∈ A, |f x|h:BddAbove shnon:s.NonemptyS:Finset XhS:sSup (Set.range fun A ↦ ∑ x ∈ A, |f x|) - ε < ∑ x ∈ S, |f x|T:Finset XhT:Disjoint T S⊢ |∑ x ∈ T, |f x|| < ε
    rw [abs_of_nonneg (by positivity)]hX:Typef:X → ℝε:ℝhε:0 < εs:Set ℝ := Set.range fun A ↦ ∑ x ∈ A, |f x|h:BddAbove shnon:s.NonemptyS:Finset XhS:sSup (Set.range fun A ↦ ∑ x ∈ A, |f x|) - ε < ∑ x ∈ S, |f x|T:Finset XhT:Disjoint T S⊢ ∑ x ∈ T, |f x| < ε
    have : ∑ x ∈ T, |f x| + ∑ x ∈ S, |f x| ≤ sSup s := byX:Typef:X → ℝ⊢ AbsConvergent' f ↔ Summable f
      apply le_csSup hX:Typef:X → ℝε:ℝhε:0 < εs:Set ℝ := Set.range fun A ↦ ∑ x ∈ A, |f x|h:BddAbove shnon:s.NonemptyS:Finset XhS:sSup (Set.range fun A ↦ ∑ x ∈ A, |f x|) - ε < ∑ x ∈ S, |f x|T:Finset XhT:Disjoint T S⊢ ∑ x ∈ T, |f x| + ∑ x ∈ S, |f x| ∈ s
      simp [s]X:Typef:X → ℝε:ℝhε:0 < εs:Set ℝ := Set.range fun A ↦ ∑ x ∈ A, |f x|h:BddAbove shnon:s.NonemptyS:Finset XhS:sSup (Set.range fun A ↦ ∑ x ∈ A, |f x|) - ε < ∑ x ∈ S, |f x|T:Finset XhT:Disjoint T S⊢ ∃ y, ∑ x ∈ y, |f x| = ∑ x ∈ T, |f x| + ∑ x ∈ S, |f x|; exact ⟨ T ∪ S, sum_union hT ⟩All goals completed! 🐙
    linarithAll goals completed! 🐙
  intro hmprX:Typef:X → ℝh:∀ (ε : ℝ), 0 < ε → ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < ε⊢ BddAbove (Set.range fun A ↦ ∑ x ∈ A, |f x|); choose S hS using h 1 (byX:Typef:X → ℝh:∀ (ε : ℝ), 0 < ε → ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < ε⊢ 0 < 1 norm_numAll goals completed! 🐙)
  rw [bddAbove_def]mprX:Typef:X → ℝh:∀ (ε : ℝ), 0 < ε → ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < εS:Finset XhS:∀ (t : Finset X), Disjoint t S → |∑ x ∈ t, |f x|| < 1⊢ ∃ x, ∀ y ∈ Set.range fun A ↦ ∑ x ∈ A, |f x|, y ≤ x
  use ∑ x ∈ S, |f x| + 1hX:Typef:X → ℝh:∀ (ε : ℝ), 0 < ε → ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < εS:Finset XhS:∀ (t : Finset X), Disjoint t S → |∑ x ∈ t, |f x|| < 1⊢ ∀ y ∈ Set.range fun A ↦ ∑ x ∈ A, |f x|, y ≤ ∑ x ∈ S, |f x| + 1; simphX:Typef:X → ℝh:∀ (ε : ℝ), 0 < ε → ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < εS:Finset XhS:∀ (t : Finset X), Disjoint t S → |∑ x ∈ t, |f x|| < 1⊢ ∀ (a : Finset X), ∑ x ∈ a, |f x| ≤ ∑ x ∈ S, |f x| + 1; intro ThX:Typef:X → ℝh:∀ (ε : ℝ), 0 < ε → ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < εS:Finset XhS:∀ (t : Finset X), Disjoint t S → |∑ x ∈ t, |f x|| < 1T:Finset X⊢ ∑ x ∈ T, |f x| ≤ ∑ x ∈ S, |f x| + 1
  calc
    _ = ∑ x ∈ (T ∩ S), |f x| + ∑ x ∈ (T \ S), |f x| := (sum_inter_add_sum_diff _ _ _).symm
    _ ≤ _ := byX:Typef:X → ℝh:∀ (ε : ℝ), 0 < ε → ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < εS:Finset XhS:∀ (t : Finset X), Disjoint t S → |∑ x ∈ t, |f x|| < 1T:Finset X⊢ ∑ x ∈ T ∩ S, |f x| + ∑ x ∈ T \ S, |f x| ≤ ∑ x ∈ S, |f x| + 1
      gcongrh₁.hX:Typef:X → ℝh:∀ (ε : ℝ), 0 < ε → ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < εS:Finset XhS:∀ (t : Finset X), Disjoint t S → |∑ x ∈ t, |f x|| < 1T:Finset X⊢ T ∩ S ⊆ Sh₂X:Typef:X → ℝh:∀ (ε : ℝ), 0 < ε → ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < εS:Finset XhS:∀ (t : Finset X), Disjoint t S → |∑ x ∈ t, |f x|| < 1T:Finset X⊢ ∑ x ∈ T \ S, |f x| ≤ 1
      .h₁.hX:Typef:X → ℝh:∀ (ε : ℝ), 0 < ε → ∃ s, ∀ (t : Finset X), Disjoint t s → |∑ x ∈ t, |f x|| < εS:Finset XhS:∀ (t : Finset X), Disjoint t S → |∑ x ∈ t, |f x|| < 1T:Finset X⊢ T ∩ S ⊆ S exact inter_subset_rightAll goals completed! 🐙
      apply le_of_lt (lt_of_abs_lt (hS _ disjoint_sdiff_self_left))All goals completed! 🐙

Maybe suitable for porting to Mathlib?

theorem Filter.Eventually.int_natCast_atTop (p: ℤ → Prop) :
  (∀ᶠ n in .atTop, p n) ↔ ∀ᶠ n:ℕ in .atTop, p ↑n := byp:ℤ → Prop⊢ (∀ᶠ (n : ℤ) in atTop, p n) ↔ ∀ᶠ (n : ℕ) in atTop, p ↑n
  refine ⟨ Eventually.natCast_atTop, ?_ ⟩p:ℤ → Prop⊢ (∀ᶠ (n : ℕ) in atTop, p ↑n) → ∀ᶠ (n : ℤ) in atTop, p n
  simp [eventually_atTop]p:ℤ → Prop⊢ ∀ (x : ℕ), (∀ (b : ℕ), x ≤ b → p ↑b) → ∃ a, ∀ (b : ℤ), a ≤ b → p b
  intro N hNp:ℤ → PropN:ℕhN:∀ (b : ℕ), N ≤ b → p ↑b⊢ ∃ a, ∀ (b : ℤ), a ≤ b → p b; use Nhp:ℤ → PropN:ℕhN:∀ (b : ℕ), N ≤ b → p ↑b⊢ ∀ (b : ℤ), ↑N ≤ b → p b; intro n hnhp:ℤ → PropN:ℕhN:∀ (b : ℕ), N ≤ b → p ↑bn:ℤhn:↑N ≤ n⊢ p n
  lift n to ℕ using (byp:ℤ → PropN:ℕhN:∀ (b : ℕ), N ≤ b → p ↑bn:ℤhn:↑N ≤ n⊢ 0 ≤ n omegaAll goals completed! 🐙)
  simp at hnhp:ℤ → PropN:ℕhN:∀ (b : ℕ), N ≤ b → p ↑bn:ℕhn:N ≤ n⊢ p ↑n; solve_by_elimAll goals completed! 🐙

theorem Filter.Tendsto.int_natCast_atTop {R:Type} (f: ℤ → R) (l: Filter R) :
atTop.Tendsto f l ↔ atTop.Tendsto (f ∘ Nat.cast) l := byR:Typef:ℤ → Rl:Filter R⊢ Tendsto f atTop l ↔ Tendsto (f ∘ Nat.cast) atTop l
  simp [tendsto_iff_eventually]R:Typef:ℤ → Rl:Filter R⊢ (∀ ⦃p : R → Prop⦄, (∀ᶠ (y : R) in l, p y) → ∃ a, ∀ (b : ℤ), a ≤ b → p (f b)) ↔
  ∀ ⦃p : R → Prop⦄, (∀ᶠ (y : R) in l, p y) → ∃ a, ∀ (b : ℕ), a ≤ b → p (f ↑b)
  peel with p hh.hR:Typef:ℤ → Rl:Filter Rp:R → Proph:∀ᶠ (y : R) in l, p y⊢ (∃ a, ∀ (b : ℤ), a ≤ b → p (f b)) ↔ ∃ a, ∀ (b : ℕ), a ≤ b → p (f ↑b)
  simp [←eventually_atTop]h.hR:Typef:ℤ → Rl:Filter Rp:R → Proph:∀ᶠ (y : R) in l, p y⊢ (∀ᶠ (x : ℤ) in atTop, p (f x)) ↔ ∀ᶠ (x : ℕ) in atTop, p (f ↑x)
  convert Eventually.int_natCast_atTop _All goals completed! 🐙

Connection with Mathlib's tsum (or Σ') operation

theorem Sum'.eq_tsum {X:Type} (f:X → ℝ) (h: AbsConvergent' f) :
  Sum' f = ∑' x, f x := byX:Typef:X → ℝh:AbsConvergent' f⊢ Sum' f = ∑' (x : X), f x
  set E := {x | f x ≠ 0}X:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}⊢ Sum' f = ∑' (x : X), f x
  obtain hE | hE := h.countable_suppinlX:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}⊢ Sum' f = ∑' (x : X), f xinrX:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:Finite ↑{x | f x ≠ 0}⊢ Sum' f = ∑' (x : X), f x
  .inlX:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}⊢ Sum' f = ∑' (x : X), f x simp [Sum']inlX:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}⊢ (Sum fun x ↦ f ↑x) = ∑' (x : X), f x
    choose g hg using hE.symminlX:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}g:ℕ → ↑{x | f x ≠ 0}hg:Bijective g⊢ (Sum fun x ↦ f ↑x) = ∑' (x : X), f x
    have : ((f ∘ Subtype.val) ∘ g:Series).absConverges := byX:Typef:X → ℝh:AbsConvergent' f⊢ Sum' f = ∑' (x : X), f x
      apply AbsConvergent.comp hgX:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}g:ℕ → ↑{x | f x ≠ 0}hg:Bijective g⊢ AbsConvergent (f ∘ Subtype.val)
      rw [←AbsConvergent'.of_countable hE]X:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}g:ℕ → ↑{x | f x ≠ 0}hg:Bijective g⊢ AbsConvergent' (f ∘ Subtype.val)
      exact h.subtype EAll goals completed! 🐙
    replace this := Sum.eq hg thisinlX:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}g:ℕ → ↑{x | f x ≠ 0}hg:Bijective gthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((f ∘ Subtype.val) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum (f ∘ Subtype.val))⊢ (Sum fun x ↦ f ↑x) = ∑' (x : X), f x
    convert convergesTo_uniq this _ using 1inl.convert_2X:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}g:ℕ → ↑{x | f x ≠ 0}hg:Bijective gthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((f ∘ Subtype.val) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum (f ∘ Subtype.val))⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then ((f ∘ Subtype.val) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo
  (∑' (x : X), f x)
    ·inl.convert_2X:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}g:ℕ → ↑{x | f x ≠ 0}hg:Bijective gthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((f ∘ Subtype.val) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum (f ∘ Subtype.val))⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then ((f ∘ Subtype.val) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo
  (∑' (x : X), f x) replace : ∑' x, f x = ∑' n, f (g n) := calc
        _ = ∑' x:E, f x := byX:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}g:ℕ → ↑{x | f x ≠ 0}hg:Bijective gthis:{ m := 0, seq := fun n ↦ if n ≥ 0 then ((f ∘ Subtype.val) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo
  (Sum (f ∘ Subtype.val))⊢ ∑' (x : X), f x = ∑' (x : ↑E), f ↑x
          exact (tsum_subtype_eq_of_support_subset (fun x hx => hx)).symmAll goals completed! 🐙
        _ = _ := (Equiv.tsum_eq (Equiv.ofBijective _ hg) _).symm
      rw [this]inl.convert_2X:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}g:ℕ → ↑{x | f x ≠ 0}hg:Bijective gthis:∑' (x : X), f x = ∑' (n : ℕ), f ↑(g n)⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then ((f ∘ Subtype.val) ∘ g) n.toNat else 0, vanish := ⋯ }.convergesTo
  (∑' (n : ℕ), f ↑(g n))
      unfold convergesToinl.convert_2X:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}g:ℕ → ↑{x | f x ≠ 0}hg:Bijective gthis:∑' (x : X), f x = ∑' (n : ℕ), f ↑(g n)⊢ Tendsto { m := 0, seq := fun n ↦ if n ≥ 0 then ((f ∘ Subtype.val) ∘ g) n.toNat else 0, vanish := ⋯ }.partial atTop
  (nhds (∑' (n : ℕ), f ↑(g n))); rw [Filter.Tendsto.int_natCast_atTop]inl.convert_2X:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}g:ℕ → ↑{x | f x ≠ 0}hg:Bijective gthis:∑' (x : X), f x = ∑' (n : ℕ), f ↑(g n)⊢ Tendsto
  ({ m := 0, seq := fun n ↦ if n ≥ 0 then ((f ∘ Subtype.val) ∘ g) n.toNat else 0, vanish := ⋯ }.partial ∘ Nat.cast)
  atTop (nhds (∑' (n : ℕ), f ↑(g n)))
      convert (Summable.tendsto_sum_tsum_nat ?_).comp (tendsto_add_atTop_nat 1) with nh.e'_3X:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}g:ℕ → ↑{x | f x ≠ 0}hg:Bijective gthis:∑' (x : X), f x = ∑' (n : ℕ), f ↑(g n)⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then ((f ∘ Subtype.val) ∘ g) n.toNat else 0, vanish := ⋯ }.partial ∘ Nat.cast =
  (fun n ↦ ∑ i ∈ range n, f ↑(g i)) ∘ fun a ↦ a + 1inl.convert_2.convert_4X:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}g:ℕ → ↑{x | f x ≠ 0}hg:Bijective gthis:∑' (x : X), f x = ∑' (n : ℕ), f ↑(g n)⊢ Summable fun n ↦ f ↑(g n)
      .h.e'_3X:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}g:ℕ → ↑{x | f x ≠ 0}hg:Bijective gthis:∑' (x : X), f x = ∑' (n : ℕ), f ↑(g n)⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then ((f ∘ Subtype.val) ∘ g) n.toNat else 0, vanish := ⋯ }.partial ∘ Nat.cast =
  (fun n ↦ ∑ i ∈ range n, f ↑(g i)) ∘ fun a ↦ a + 1 ext Nh.e'_3.hX:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}g:ℕ → ↑{x | f x ≠ 0}hg:Bijective gthis:∑' (x : X), f x = ∑' (n : ℕ), f ↑(g n)N:ℕ⊢ ({ m := 0, seq := fun n ↦ if n ≥ 0 then ((f ∘ Subtype.val) ∘ g) n.toNat else 0, vanish := ⋯ }.partial ∘ Nat.cast) N =
  ((fun n ↦ ∑ i ∈ range n, f ↑(g i)) ∘ fun a ↦ a + 1) N; simp [Series.partial, Nat.range_succ_eq_Icc_zero]All goals completed! 🐙
      rw [AbsConvergent'.iff_Summable] at hinl.convert_2.convert_4X:Typef:X → ℝh:Summable fE:Set X := {x | f x ≠ 0}hE:CountablyInfinite ↑{x | f x ≠ 0}g:ℕ → ↑{x | f x ≠ 0}hg:Bijective gthis:∑' (x : X), f x = ∑' (n : ℕ), f ↑(g n)⊢ Summable fun n ↦ f ↑(g n)
      exact h.comp_injective (i := Subtype.val ∘ g) (Subtype.val_injective.comp hg.1)All goals completed! 🐙
  rw [of_finsupp (A := E.toFinite.toFinset)]inrX:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:Finite ↑{x | f x ≠ 0}⊢ ∑ x ∈ ⋯.toFinset, f x = ∑' (x : X), f xinrX:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:Finite ↑{x | f x ≠ 0}⊢ ∀ x ∉ ⋯.toFinset, f x = 0; symminrX:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:Finite ↑{x | f x ≠ 0}⊢ ∑' (x : X), f x = ∑ x ∈ ⋯.toFinset, f xinrX:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:Finite ↑{x | f x ≠ 0}⊢ ∀ x ∉ ⋯.toFinset, f x = 0; apply tsum_eq_suminr.hfX:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:Finite ↑{x | f x ≠ 0}⊢ ∀ b ∉ ⋯.toFinset, f b = 0inrX:Typef:X → ℝh:AbsConvergent' fE:Set X := {x | f x ≠ 0}hE:Finite ↑{x | f x ≠ 0}⊢ ∀ x ∉ ⋯.toFinset, f x = 0
  all_goals simp [E]All goals completed! 🐙

Proposition 8.2.6 (a) (Absolutely convergent series laws) / Exercise 8.2.3

theorem declaration uses `sorry`Sum'.add {X:Type} {f g:X → ℝ} (hf: AbsConvergent' f) (hg: AbsConvergent' g) :
  AbsConvergent' (f+g) ∧ Sum' (f + g) = Sum' f + Sum' g := byX:Typef:X → ℝg:X → ℝhf:AbsConvergent' fhg:AbsConvergent' g⊢ AbsConvergent' (f + g) ∧ Sum' (f + g) = Sum' f + Sum' g
  sorryAll goals completed! 🐙

Proposition 8.2.6 (b) (Absolutely convergent series laws) / Exercise 8.2.3

theorem declaration uses `sorry`Sum'.smul {X:Type} {f:X → ℝ} (hf: AbsConvergent' f) (c: ℝ) :
  AbsConvergent' (c • f) ∧ Sum' (c • f) = c * Sum' f := byX:Typef:X → ℝhf:AbsConvergent' fc:ℝ⊢ AbsConvergent' (c • f) ∧ Sum' (c • f) = c * Sum' f
  sorryAll goals completed! 🐙

This law is not explicitly stated in Proposition 8.2.6, but follows easily from parts (a) and (b).

theorem Sum'.sub {X:Type} {f g:X → ℝ} (hf: AbsConvergent' f) (hg: AbsConvergent' g) :
  AbsConvergent' (f-g) ∧ Sum' (f - g) = Sum' f - Sum' g := byX:Typef:X → ℝg:X → ℝhf:AbsConvergent' fhg:AbsConvergent' g⊢ AbsConvergent' (f - g) ∧ Sum' (f - g) = Sum' f - Sum' g
  convert add hf (smul hg (-1)).1 using 2h.e'_1.h.e'_2X:Typef:X → ℝg:X → ℝhf:AbsConvergent' fhg:AbsConvergent' g⊢ f - g = f + -1 • gh.e'_2.h.e'_2X:Typef:X → ℝg:X → ℝhf:AbsConvergent' fhg:AbsConvergent' g⊢ Sum' (f - g) = Sum' (f + -1 • g)h.e'_2.h.e'_3X:Typef:X → ℝg:X → ℝhf:AbsConvergent' fhg:AbsConvergent' g⊢ Sum' f - Sum' g = Sum' f + Sum' (-1 • g)
  .h.e'_1.h.e'_2X:Typef:X → ℝg:X → ℝhf:AbsConvergent' fhg:AbsConvergent' g⊢ f - g = f + -1 • g simph.e'_1.h.e'_2X:Typef:X → ℝg:X → ℝhf:AbsConvergent' fhg:AbsConvergent' g⊢ f - g = f + -g; abelAll goals completed! 🐙
  .h.e'_2.h.e'_2X:Typef:X → ℝg:X → ℝhf:AbsConvergent' fhg:AbsConvergent' g⊢ Sum' (f - g) = Sum' (f + -1 • g) congrh.e'_2.h.e'_2.e_fX:Typef:X → ℝg:X → ℝhf:AbsConvergent' fhg:AbsConvergent' g⊢ f - g = f + -1 • g; simph.e'_2.h.e'_2.e_fX:Typef:X → ℝg:X → ℝhf:AbsConvergent' fhg:AbsConvergent' g⊢ f - g = f + -g; abelAll goals completed! 🐙
  rw [(smul hg (-1)).2]h.e'_2.h.e'_3X:Typef:X → ℝg:X → ℝhf:AbsConvergent' fhg:AbsConvergent' g⊢ Sum' f - Sum' g = Sum' f + -1 * Sum' g; ringAll goals completed! 🐙

Proposition 8.2.6 (c) (Absolutely convergent series laws) / Exercise 8.2.3. The first part of this proposition has been moved to AbsConvergent'.subtype.

theorem declaration uses `sorry`Sum'.of_disjoint_union {X:Type} {f:X → ℝ} (hf: AbsConvergent' f) {X₁ X₂ : Set X} (hdisj: Disjoint X₁ X₂):
  Sum' (fun x: (X₁ ∪ X₂: Set X) ↦ f x) = Sum' (fun x : X₁ ↦ f x) + Sum' (fun x : X₂ ↦ f x) := byX:Typef:X → ℝhf:AbsConvergent' fX₁:Set XX₂:Set Xhdisj:Disjoint X₁ X₂⊢ (Sum' fun x ↦ f ↑x) = (Sum' fun x ↦ f ↑x) + Sum' fun x ↦ f ↑x
  sorryAll goals completed! 🐙

This technical claim, the analogue of tsum_univ, is required due to the way Mathlib handles sets.

theorem declaration uses `sorry`Sum'.of_univ {X:Type} {f:X → ℝ} (hf: AbsConvergent' f) :
  Sum' (fun x: (.univ : Set X) ↦ f x) = Sum' f := byX:Typef:X → ℝhf:AbsConvergent' f⊢ (Sum' fun x ↦ f ↑x) = Sum' f
  sorryAll goals completed! 🐙

theorem declaration uses `sorry`Sum'.of_comp {X Y:Type} {f:X → ℝ} (hf: AbsConvergent' f) {φ: Y → X}
  (hφ: Function.Bijective φ) :
  AbsConvergent' (f ∘ φ) ∧ Sum' f = Sum' (f ∘ φ) := byX:TypeY:Typef:X → ℝhf:AbsConvergent' fφ:Y → Xhφ:Bijective φ⊢ AbsConvergent' (f ∘ φ) ∧ Sum' f = Sum' (f ∘ φ)
  sorryAll goals completed! 🐙

Lemma 8.2.7 / Exercise 8.2.4

theorem declaration uses `sorry`divergent_parts_of_divergent {a: ℕ → ℝ} (ha: (a:Series).converges)
  (ha': ¬ (a:Series).absConverges) :
  ¬ AbsConvergent (fun n : {n | a n ≥ 0} ↦ a n) ∧ ¬ AbsConvergent (fun n : {n | a n < 0} ↦ a n)
  := bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConverges⊢ (¬AbsConvergent fun n ↦ a ↑n) ∧ ¬AbsConvergent fun n ↦ a ↑n
  sorryAll goals completed! 🐙

Theorem 8.2.8 (Riemann rearrangement theorem) / Exercise 8.2.5

theorem declaration uses `sorry`permute_convergesTo_of_divergent {a: ℕ → ℝ} (ha: (a:Series).converges)
  (ha': ¬ (a:Series).absConverges) (L:ℝ) :
  ∃ f : ℕ → ℕ, Bijective f ∧ (a ∘ f:Series).convergesTo L
  := bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝ⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L
  -- This proof is written to follow the structure of the original text.
  choose h1 h2 using divergent_parts_of_divergent ha ha'a:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝh1:¬AbsConvergent fun n ↦ a ↑nh2:¬AbsConvergent fun n ↦ a ↑n⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L
  set A_plus := { n | a n ≥ 0 }a:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝh2:¬AbsConvergent fun n ↦ a ↑nA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑n⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L
  set A_minus := {n | a n < 0 }a:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑n⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L
  have hdisj : Disjoint A_plus A_minus := bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝ⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L
    rw [Set.disjoint_iff_inter_eq_empty]a:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑n⊢ A_plus ∩ A_minus = ∅; extha:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑nx✝:ℕ⊢ x✝ ∈ A_plus ∩ A_minus ↔ x✝ ∈ ∅; simp [A_plus, A_minus]All goals completed! 🐙
  have hunion : A_plus ∪ A_minus = .univ := bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝ⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L
    extha:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑nhdisj:Disjoint A_plus A_minusx✝:ℕ⊢ x✝ ∈ A_plus ∪ A_minus ↔ x✝ ∈ Set.univ; simp [A_plus, A_minus]ha:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑nhdisj:Disjoint A_plus A_minusx✝:ℕ⊢ 0 ≤ a x✝ ∨ a x✝ < 0; grindAll goals completed! 🐙
  have hA_plus_inf : Infinite A_plus := sorrya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑nhdisj:Disjoint A_plus A_minushunion:A_plus ∪ A_minus = Set.univhA_plus_inf:Infinite ↑A_plus⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L
  have hA_minus_inf : Infinite A_minus := sorrya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑nhdisj:Disjoint A_plus A_minushunion:A_plus ∪ A_minus = Set.univhA_plus_inf:Infinite ↑A_plushA_minus_inf:Infinite ↑A_minus⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L
  obtain ⟨ a_plus, ha_plus_bij, ha_plus_mono ⟩ := (Nat.monotone_enum_of_infinite A_plus).existsa:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑nhdisj:Disjoint A_plus A_minushunion:A_plus ∪ A_minus = Set.univhA_plus_inf:Infinite ↑A_plushA_minus_inf:Infinite ↑A_minusa_plus:ℕ → ↑A_plusha_plus_bij:Bijective a_plusha_plus_mono:StrictMono a_plus⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L
  obtain ⟨ a_minus, ha_minus_bij, ha_minus_mono ⟩ := (Nat.monotone_enum_of_infinite A_minus).existsa:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑nhdisj:Disjoint A_plus A_minushunion:A_plus ∪ A_minus = Set.univhA_plus_inf:Infinite ↑A_plushA_minus_inf:Infinite ↑A_minusa_plus:ℕ → ↑A_plusha_plus_bij:Bijective a_plusha_plus_mono:StrictMono a_plusa_minus:ℕ → ↑A_minusha_minus_bij:Bijective a_minusha_minus_mono:StrictMono a_minus⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L
  let F : (n : ℕ) → ((m : ℕ) → m < n → ℕ) → ℕ :=
    fun j n' ↦ if ∑ i:Fin j, a (n' i (bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑nhdisj:Disjoint A_plus A_minushunion:A_plus ∪ A_minus = Set.univhA_plus_inf:Infinite ↑A_plushA_minus_inf:Infinite ↑A_minusa_plus:ℕ → ↑A_plusha_plus_bij:Bijective a_plusha_plus_mono:StrictMono a_plusa_minus:ℕ → ↑A_minusha_minus_bij:Bijective a_minusha_minus_mono:StrictMono a_minusj:ℕn':(m : ℕ) → m < j → ℕi:Fin j⊢ ↑i < j simpAll goals completed! 🐙)) < L then
      Nat.min { n ∈ A_plus | ∀ i:Fin j, n ≠ n' i (bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑nhdisj:Disjoint A_plus A_minushunion:A_plus ∪ A_minus = Set.univhA_plus_inf:Infinite ↑A_plushA_minus_inf:Infinite ↑A_minusa_plus:ℕ → ↑A_plusha_plus_bij:Bijective a_plusha_plus_mono:StrictMono a_plusa_minus:ℕ → ↑A_minusha_minus_bij:Bijective a_minusha_minus_mono:StrictMono a_minusj:ℕn':(m : ℕ) → m < j → ℕn:ℕi:Fin j⊢ ↑i < j simpAll goals completed! 🐙) }
    else
      Nat.min { n ∈ A_minus | ∀ i:Fin j, n ≠ n' i (bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑nhdisj:Disjoint A_plus A_minushunion:A_plus ∪ A_minus = Set.univhA_plus_inf:Infinite ↑A_plushA_minus_inf:Infinite ↑A_minusa_plus:ℕ → ↑A_plusha_plus_bij:Bijective a_plusha_plus_mono:StrictMono a_plusa_minus:ℕ → ↑A_minusha_minus_bij:Bijective a_minusha_minus_mono:StrictMono a_minusj:ℕn':(m : ℕ) → m < j → ℕn:ℕi:Fin j⊢ ↑i < j simpAll goals completed! 🐙) }
  let n' : ℕ → ℕ := Nat.strongRec Fa:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑nhdisj:Disjoint A_plus A_minushunion:A_plus ∪ A_minus = Set.univhA_plus_inf:Infinite ↑A_plushA_minus_inf:Infinite ↑A_minusa_plus:ℕ → ↑A_plusha_plus_bij:Bijective a_plusha_plus_mono:StrictMono a_plusa_minus:ℕ → ↑A_minusha_minus_bij:Bijective a_minusha_minus_mono:StrictMono a_minusF:(n : ℕ) → ((m : ℕ) → m < n → ℕ) → ℕ := 
  fun j n' ↦
    if ∑ i, a (n' ↑i ⋯) < L then Nat.min {n | n ∈ A_plus ∧ ∀ (i : Fin j), n ≠ n' ↑i ⋯}
    else Nat.min {n | n ∈ A_minus ∧ ∀ (i : Fin j), n ≠ n' ↑i ⋯}n':ℕ → ℕ := fun t ↦ Nat.strongRec F t⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L
  have hn' (j:ℕ) : n' j = if ∑ i:Fin j, a (n' i) < L then
      Nat.min { n ∈ A_plus | ∀ i:Fin j, n ≠ n' i }
    else
      Nat.min { n ∈ A_minus | ∀ i:Fin j, n ≠ n' i }
    := Nat.strongRec.eq_def _ ja:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑nhdisj:Disjoint A_plus A_minushunion:A_plus ∪ A_minus = Set.univhA_plus_inf:Infinite ↑A_plushA_minus_inf:Infinite ↑A_minusa_plus:ℕ → ↑A_plusha_plus_bij:Bijective a_plusha_plus_mono:StrictMono a_plusa_minus:ℕ → ↑A_minusha_minus_bij:Bijective a_minusha_minus_mono:StrictMono a_minusF:(n : ℕ) → ((m : ℕ) → m < n → ℕ) → ℕ := 
  fun j n' ↦
    if ∑ i, a (n' ↑i ⋯) < L then Nat.min {n | n ∈ A_plus ∧ ∀ (i : Fin j), n ≠ n' ↑i ⋯}
    else Nat.min {n | n ∈ A_minus ∧ ∀ (i : Fin j), n ≠ n' ↑i ⋯}n':ℕ → ℕ := fun t ↦ Nat.strongRec F thn':∀ (j : ℕ),
  n' j =
    if ∑ i, a (n' ↑i) < L then Nat.min {n | n ∈ A_plus ∧ ∀ (i : Fin j), n ≠ n' ↑i}
    else Nat.min {n | n ∈ A_minus ∧ ∀ (i : Fin j), n ≠ n' ↑i}⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L
  have hn'_plus_inf (j:ℕ) : Infinite { n ∈ A_plus | ∀ i:Fin j, n ≠ n' i } := bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝ⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L sorryAll goals completed! 🐙
  have hn'_minus_inf (j:ℕ) : Infinite { n ∈ A_minus | ∀ i:Fin j, n ≠ n' i } := bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝ⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L sorryAll goals completed! 🐙
  have hn'_inj : Injective n' := bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝ⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L sorryAll goals completed! 🐙
  have h_case_I : Infinite { j | ∑ i:Fin j, a (n' i) < L } := bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝ⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L sorryAll goals completed! 🐙
  have h_case_II : Infinite { j | ∑ i:Fin j, a (n' i) ≥ L } := bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝ⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L sorryAll goals completed! 🐙
  have hn'_surj : Surjective n' := bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝ⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L sorryAll goals completed! 🐙
  have hconv : atTop.Tendsto (a ∘ n') (nhds 0) := bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝ⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L sorryAll goals completed! 🐙
  have hsum : (a ∘ n':Series).convergesTo L := bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝ⊢ ∃ f, Bijective f ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.convergesTo L sorryAll goals completed! 🐙
  use n'ha:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑nhdisj:Disjoint A_plus A_minushunion:A_plus ∪ A_minus = Set.univhA_plus_inf:Infinite ↑A_plushA_minus_inf:Infinite ↑A_minusa_plus:ℕ → ↑A_plusha_plus_bij:Bijective a_plusha_plus_mono:StrictMono a_plusa_minus:ℕ → ↑A_minusha_minus_bij:Bijective a_minusha_minus_mono:StrictMono a_minusF:(n : ℕ) → ((m : ℕ) → m < n → ℕ) → ℕ := 
  fun j n' ↦
    if ∑ i, a (n' ↑i ⋯) < L then Nat.min {n | n ∈ A_plus ∧ ∀ (i : Fin j), n ≠ n' ↑i ⋯}
    else Nat.min {n | n ∈ A_minus ∧ ∀ (i : Fin j), n ≠ n' ↑i ⋯}n':ℕ → ℕ := fun t ↦ Nat.strongRec F thn':∀ (j : ℕ),
  n' j =
    if ∑ i, a (n' ↑i) < L then Nat.min {n | n ∈ A_plus ∧ ∀ (i : Fin j), n ≠ n' ↑i}
    else Nat.min {n | n ∈ A_minus ∧ ∀ (i : Fin j), n ≠ n' ↑i}hn'_plus_inf:∀ (j : ℕ), Infinite ↑{n | n ∈ A_plus ∧ ∀ (i : Fin j), n ≠ n' ↑i}hn'_minus_inf:∀ (j : ℕ), Infinite ↑{n | n ∈ A_minus ∧ ∀ (i : Fin j), n ≠ n' ↑i}hn'_inj:Injective n'h_case_I:Infinite ↑{j | ∑ i, a (n' ↑i) < L}h_case_II:Infinite ↑{j | ∑ i, a (n' ↑i) ≥ L}hn'_surj:Surjective n'hconv:Tendsto (a ∘ n') atTop (nhds 0)hsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ n') n.toNat else 0, vanish := ⋯ }.convergesTo L⊢ Bijective n' ∧ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ n') n.toNat else 0, vanish := ⋯ }.convergesTo L
  refine ⟨ ⟨ hn'_inj, hn'_surj ⟩, ?_ ⟩ha:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesL:ℝA_plus:Set ℕ := {n | a n ≥ 0}h1:¬AbsConvergent fun n ↦ a ↑nA_minus:Set ℕ := {n | a n < 0}h2:¬AbsConvergent fun n ↦ a ↑nhdisj:Disjoint A_plus A_minushunion:A_plus ∪ A_minus = Set.univhA_plus_inf:Infinite ↑A_plushA_minus_inf:Infinite ↑A_minusa_plus:ℕ → ↑A_plusha_plus_bij:Bijective a_plusha_plus_mono:StrictMono a_plusa_minus:ℕ → ↑A_minusha_minus_bij:Bijective a_minusha_minus_mono:StrictMono a_minusF:(n : ℕ) → ((m : ℕ) → m < n → ℕ) → ℕ := 
  fun j n' ↦
    if ∑ i, a (n' ↑i ⋯) < L then Nat.min {n | n ∈ A_plus ∧ ∀ (i : Fin j), n ≠ n' ↑i ⋯}
    else Nat.min {n | n ∈ A_minus ∧ ∀ (i : Fin j), n ≠ n' ↑i ⋯}n':ℕ → ℕ := fun t ↦ Nat.strongRec F thn':∀ (j : ℕ),
  n' j =
    if ∑ i, a (n' ↑i) < L then Nat.min {n | n ∈ A_plus ∧ ∀ (i : Fin j), n ≠ n' ↑i}
    else Nat.min {n | n ∈ A_minus ∧ ∀ (i : Fin j), n ≠ n' ↑i}hn'_plus_inf:∀ (j : ℕ), Infinite ↑{n | n ∈ A_plus ∧ ∀ (i : Fin j), n ≠ n' ↑i}hn'_minus_inf:∀ (j : ℕ), Infinite ↑{n | n ∈ A_minus ∧ ∀ (i : Fin j), n ≠ n' ↑i}hn'_inj:Injective n'h_case_I:Infinite ↑{j | ∑ i, a (n' ↑i) < L}h_case_II:Infinite ↑{j | ∑ i, a (n' ↑i) ≥ L}hn'_surj:Surjective n'hconv:Tendsto (a ∘ n') atTop (nhds 0)hsum:{ m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ n') n.toNat else 0, vanish := ⋯ }.convergesTo L⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ n') n.toNat else 0, vanish := ⋯ }.convergesTo L; convert hsumAll goals completed! 🐙

Exercise 8.2.6

theorem declaration uses `sorry`permute_diverges_of_divergent {a: ℕ → ℝ} (ha: (a:Series).converges)
  (ha': ¬ (a:Series).absConverges)  :
  ∃ f : ℕ → ℕ,  Bijective f ∧ atTop.Tendsto (fun N ↦ ((a ∘ f:Series).partial N : EReal)) (nhds ⊤) := bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConverges⊢ ∃ f,
  Bijective f ∧
    Tendsto (fun N ↦ ↑({ m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.partial N)) atTop
      (nhds ⊤)
  sorryAll goals completed! 🐙

theorem declaration uses `sorry`permute_diverges_of_divergent' {a: ℕ → ℝ} (ha: (a:Series).converges)
  (ha': ¬ (a:Series).absConverges)  :
  ∃ f : ℕ → ℕ,  Bijective f ∧ atTop.Tendsto (fun N ↦ ((a ∘ f:Series).partial N : EReal)) (nhds ⊥) := bya:ℕ → ℝha:{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesha':¬{ m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConverges⊢ ∃ f,
  Bijective f ∧
    Tendsto (fun N ↦ ↑({ m := 0, seq := fun n ↦ if n ≥ 0 then (a ∘ f) n.toNat else 0, vanish := ⋯ }.partial N)) atTop
      (nhds ⊥)
  sorryAll goals completed! 🐙

end Chapter8