-- Mathlib/Analysis/Normed/Group/Tannery.lean
/-- **Tannery's theorem**: topological sums commute with termwise limits, when the norms of the
summands are eventually uniformly bounded by a summable function.

(This is the special case of the Lebesgue dominated convergence theorem for the counting measure
on a discrete set. However, we prove it under somewhat weaker assumptions than the general
measure-theoretic result, e.g. `G` is not assumed to be an `ℝ`-vector space or second countable,
and the limit is along an arbitrary filter rather than `atTop ℕ`.)

See also:
* `MeasureTheory.tendsto_integral_of_dominated_convergence` (for general integrals, but
  with more assumptions on `G`)
* `continuous_tsum` (continuity of infinite sums in a parameter)
-/
lemma tendsto_tsum_of_dominated_convergence {α β G : Type*} {𝓕 : Filter α}
    [NormedAddCommGroup G] [CompleteSpace G]
    {f : α → β → G} {g : β → G} {bound : β → ℝ} (h_sum : Summable bound)
    (hab : ∀ k : β, Tendsto (f · k) 𝓕 (𝓝 (g k)))
    (h_bound : ∀ᶠ n in 𝓕, ∀ k, ‖f n k‖ ≤ bound k) :
    Tendsto (∑' k, f · k) 𝓕 (𝓝 (∑' k, g k))

Proof state:
case inr.inr.intro
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x => ‖g x‖
h_suma : ∀ᶠ (n : α) in 𝓕, Summable fun x => ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
h1 : ∑' (k : ↑(↑T)ᶜ), bound ↑k < ε / 3
h2 : Tendsto (fun x => ∑ k ∈ T, f x k) 𝓕 (𝓝 (T.sum g))
⊢ ∀ᶠ (x : α) in 𝓕, dist (∑' (k : β), f x k) (∑' (k : β), g k) < ε

Tactic:
rw [Metric.tendsto_nhds] at h2

Result:
case inr.inr.intro
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x => ‖g x‖
h_suma : ∀ᶠ (n : α) in 𝓕, Summable fun x => ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
h1 : ∑' (k : ↑(↑T)ᶜ), bound ↑k < ε / 3
h2 : ∀ ε > 0, ∀ᶠ (x : α) in 𝓕, dist (∑ k ∈ T, f x k) (T.sum g) < ε
⊢ ∀ᶠ (x : α) in 𝓕, dist (∑' (k : β), f x k) (∑' (k : β), g k) < ε

Proof state:
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x => ‖g x‖
h_suma : ∀ᶠ (n : α) in 𝓕, Summable fun x => ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : ∀ ε > 0, ∀ᶠ (x : Finset β) in atTop, dist (∑ b ∈ x, bound b) S < ε
⊢ ε / 3 > 0

Tactic:
positivity

Result:
no goals

Proof state:
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x => ‖g x‖
h_suma : ∀ᶠ (n : α) in 𝓕, Summable fun x => ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : ∀ ε > 0, ∀ᶠ (x : Finset β) in atTop, dist (∑ b ∈ x, bound b) S < ε
⊢ ∃ T, dist (∑ b ∈ T, bound b) S < ε / 3

Tactic:
exact Eventually.exists <| hS _ (by positivity)

Result:
no goals

Proof state:
case h
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound✝ : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x => ‖g x‖
h_suma✝ : ∀ᶠ (n : α) in 𝓕, Summable fun x => ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
h1 : ∑' (k : ↑(↑T)ᶜ), bound ↑k < ε / 3
h2 : ∀ ε > 0, ∀ᶠ (x : α) in 𝓕, dist (∑ k ∈ T, f x k) (T.sum g) < ε
n : α
hn : dist (∑ k ∈ T, f n k) (T.sum g) < ε / 3
h_suma : Summable fun x => ‖f n x‖
h_bound : ∀ (k : β), ‖f n k‖ ≤ bound k
⊢ dist (∑' (k : β), f n k) (∑' (k : β), g k) < ε

Tactic:
rw [dist_eq_norm, ← tsum_sub h_suma.of_norm h_sumg.of_norm,
  ← sum_add_tsum_compl (s := T) (h_suma.of_norm.sub h_sumg.of_norm),
  (by ring : ε = ε / 3 + (ε / 3 + ε / 3))]

Result:
case h
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound✝ : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x => ‖g x‖
h_suma✝ : ∀ᶠ (n : α) in 𝓕, Summable fun x => ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
h1 : ∑' (k : ↑(↑T)ᶜ), bound ↑k < ε / 3
h2 : ∀ ε > 0, ∀ᶠ (x : α) in 𝓕, dist (∑ k ∈ T, f x k) (T.sum g) < ε
n : α
hn : dist (∑ k ∈ T, f n k) (T.sum g) < ε / 3
h_suma : Summable fun x => ‖f n x‖
h_bound : ∀ (k : β), ‖f n k‖ ≤ bound k
⊢ ‖∑ x ∈ T, (f n x - g x) + ∑' (x : ↑(↑T)ᶜ), (f n ↑x - g ↑x)‖ < ε / 3 + (ε / 3 + ε / 3)

Proof state:
case inr.inr
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x => ‖g x‖
h_suma : ∀ᶠ (n : α) in 𝓕, Summable fun x => ‖f n x‖
⊢ ∀ ε > 0, ∀ᶠ (x : α) in 𝓕, dist (∑' (k : β), f x k) (∑' (k : β), g k) < ε

Tactic:
intro ε hε

Result:
case inr.inr
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x => ‖g x‖
h_suma : ∀ᶠ (n : α) in 𝓕, Summable fun x => ‖f n x‖
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (x : α) in 𝓕, dist (∑' (k : β), f x k) (∑' (k : β), g k) < ε

Proof state:
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x => ‖g x‖
h_suma : ∀ᶠ (n : α) in 𝓕, Summable fun x => ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
⊢ ∃ T, dist (∑ b ∈ T, bound b) S < ε / 3

Tactic:
rw [HasSum, Metric.tendsto_nhds] at hS

Result:
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x => ‖g x‖
h_suma : ∀ᶠ (n : α) in 𝓕, Summable fun x => ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : ∀ ε > 0, ∀ᶠ (x : Finset β) in atTop, dist (∑ b ∈ x, bound b) S < ε
⊢ ∃ T, dist (∑ b ∈ T, bound b) S < ε / 3