-- Module: Mathlib.Analysis.SpecialFunctions.Log.Summable
Goal:
ι : Type u_1
R : Type u_2
inst : NormedCommRing R
inst_1 : NormOneClass R
f : ι → R
inst_2 : CompleteSpace R
hf : Summable (α := ℝ) fun (i : ι) ↦ ‖f i‖
refine :
  ∀ (ε : ℝ) (hε : ε > 0),
    ∃ (t : Set R),
      Membership.mem (γ := Filter R)
          (Filter.map (fun (s : Finset ι) ↦ ∏ b ∈ s, (fun (i : ι) ↦ 1 + f i) b) (unconditional ι).filter) t ∧
        ∀ x ∈ t, ∀ y ∈ t, dist x y < ε
refine₁ :
  ∀ (ε : ℝ) (hε : ε > 0) (r₁ : ℝ) (hr₁ : r₁ > 0) (s₁ : Finset ι)
    (hs₁ : ∀ (t : Finset ι), s₁ ⊆ t → ∏ i ∈ t, ‖1 + f i‖ ≤ r₁),
    ∃ (t : Set R),
      Membership.mem (γ := Filter R)
          (Filter.map (fun (s : Finset ι) ↦ ∏ b ∈ s, (fun (i : ι) ↦ 1 + f i) b) (unconditional ι).filter) t ∧
        ∀ x ∈ t, ∀ y ∈ t, dist x y < ε
refine₂ :
  ∀ (ε : ℝ) (hε : ε > 0) (r₁ : ℝ) (hr₁ : r₁ > 0) (s₁ : Finset ι)
    (hs₁ : ∀ (t : Finset ι), s₁ ⊆ t → ∏ i ∈ t, ‖1 + f i‖ ≤ r₁) (s₂ : Finset ι)
    (hs₂ : ∀ (t : Finset ι), Disjoint t s₂ → ‖HSub.hSub (α := R) (∏ i ∈ t, (1 + f i)) 1‖ < ε / (2 * r₁)),
    ∃ (t : Set R),
      Membership.mem (γ := Filter R)
          (Filter.map (fun (s : Finset ι) ↦ ∏ b ∈ s, (fun (i : ι) ↦ 1 + f i) b) (unconditional ι).filter) t ∧
        ∀ x ∈ t, ∀ y ∈ t, dist x y < ε
refine₃ :
  ∀ (ε : ℝ) (hε : ε > 0) (r₁ : ℝ) (hr₁ : r₁ > 0) (s₁ : Finset ι)
    (hs₁ : ∀ (t : Finset ι), s₁ ⊆ t → ∏ i ∈ t, ‖1 + f i‖ ≤ r₁) (s₂ : Finset ι)
    (hs₂ : ∀ (t : Finset ι), Disjoint t s₂ → ‖HSub.hSub (α := R) (∏ i ∈ t, (1 + f i)) 1‖ < ε / (2 * r₁)),
    ∃ (t : Set R), (∃ (a : Finset ι), ∀ (b : Finset ι), a ⊆ b → ∏ x ∈ b, (1 + f x) ∈ t) ∧ ∀ x ∈ t, ∀ y ∈ t, dist x y < ε
refine₄ :
  ∀ (ε : ℝ) (hε : ε > 0) (r₁ : ℝ) (hr₁ : r₁ > 0) (s₁ : Finset ι)
    (hs₁ : ∀ (t : Finset ι), s₁ ⊆ t → ∏ i ∈ t, ‖1 + f i‖ ≤ r₁) (s₂ : Finset ι)
    (hs₂ : ∀ (t : Finset ι), Disjoint t s₂ → ‖HSub.hSub (α := R) (∏ i ∈ t, (1 + f i)) 1‖ < ε / (2 * r₁)),
    ∃ (t : Set R), (∃ (a : Finset ι), ∀ (b : Finset ι), a ⊆ b → ∏ x ∈ b, (1 + f x) ∈ t) ∧ ∀ x ∈ t, ∀ y ∈ t, dist x y < ε
refine_1 :
  ∀ (ε : ℝ) (hε : ε > 0) (r₁ : ℝ) (hr₁ : r₁ > 0) (s₁ : Finset ι)
    (hs₁ : ∀ (t : Finset ι), s₁ ⊆ t → ∏ i ∈ t, ‖1 + f i‖ ≤ r₁) (s₂ : Finset ι)
    (hs₂ : ∀ (t : Finset ι), Disjoint t s₂ → ‖HSub.hSub (α := R) (∏ i ∈ t, (1 + f i)) 1‖ < ε / (2 * r₁)),
    let s := s₁ ∪ s₂;
    ∀ (b : Finset ι) (hb : s ⊆ b),
      Membership.mem (γ := Set R) (Metric.ball (∏ i ∈ s, (1 + f i)) (ε / 2)) (∏ x ∈ b, (1 + f x))
refine_2 refine_2₁ :
  ∀ (ε : ℝ) (hε : ε > 0) (r₁ : ℝ) (hr₁ : r₁ > 0) (s₁ : Finset ι)
    (hs₁ : ∀ (t : Finset ι), s₁ ⊆ t → ∏ i ∈ t, ‖1 + f i‖ ≤ r₁) (s₂ : Finset ι)
    (hs₂ : ∀ (t : Finset ι), Disjoint t s₂ → ‖HSub.hSub (α := R) (∏ i ∈ t, (1 + f i)) 1‖ < ε / (2 * r₁)),
    let s := s₁ ∪ s₂;
    ∀ (x : R),
      Membership.mem (γ := Set R) (Metric.ball (∏ i ∈ s, (1 + f i)) (ε / 2)) x →
        ∀ (y : R), Membership.mem (γ := Set R) (Metric.ball (∏ i ∈ s, (1 + f i)) (ε / 2)) y → dist x y < ε
⊢ Multipliable.{u_2, u_1} (α := R) fun (i : ι) ↦ 1 + f i

Tactic:
classical
  refine CompleteSpace.complete <| Metric.cauchy_iff.mpr ⟨by infer_instance, fun ε hε ↦ ?_⟩
  obtain ⟨r₁, hr₁, s₁, hs₁⟩ :=
    (multipliable_norm_one_add_of_summable_norm hf).eventually_bounded_finsetProd
  obtain ⟨s₂, hs₂⟩ := prod_vanishing_of_summable_norm hf (show 0 < ε / (2 * r₁) by positivity)
  simp only [unconditional, Filter.mem_map, mem_atTop_sets, ge_iff_le, le_eq_subset,
    Set.mem_preimage]
  let s := s₁ ∪ s₂
  -- The idea here is that if `s` is a large enough finset, then the product over `s` is bounded
  -- by some `r`, and the product over finsets disjoint from `s` is within `ε / (2 * r)` of 1.
  -- From this it follows that the products over any two finsets containing `s` are within `ε` of
  -- each other.
  -- Here `s₁ ⊆ s` guarantees that the product over `s` is bounded, and `s₂ ⊆ s` guarantees that
  -- the product over terms not in `s` is small.
  refine ⟨Metric.ball (∏ i ∈ s, (1 + f i)) (ε / 2), ⟨s, fun b hb ↦ ?_⟩, ?_⟩
  · rw [← union_sdiff_of_subset hb, prod_union sdiff_disjoint.symm,
      Metric.mem_ball, dist_eq_norm_sub, ← mul_sub_one,
      show ε / 2 = r₁ * (ε / (2 * r₁)) by field]
    apply (norm_mul_le _ _).trans_lt
    refine lt_of_le_of_lt (b := r₁ * ‖∏ x ∈ b \ s, (1 + f x) - 1‖) ?_ ?_
    · refine mul_le_mul_of_nonneg_right ?_ (norm_nonneg _)
      exact (Finset.norm_prod_le _ _).trans (hs₁ _ subset_union_left)
    · refine mul_lt_mul_of_pos_left (hs₂ _ ?_) hr₁
      simp [s, sdiff_union_distrib, disjoint_iff_inter_eq_empty]
  · intro x hx y hy
    exact (dist_triangle_right _ _ (∏ i ∈ s, (1 + f i))).trans_lt (add_halves ε ▸ add_lt_add hx hy)

Premises used: Iff.mpr, dist_eq_norm_sub, ge_iff_le, Filter.mem_atTop_sets, multipliable_norm_one_add_of_summable_norm, Finset.prod_union, mul_sub_one, Metric.mem_ball, mul_le_mul_of_nonneg_right, Finset.disjoint_iff_inter_eq_empty, Disjoint.symm, Metric.cauchy_iff, CompleteSpace.complete, And.intro, Finset.le_eq_subset, norm_mul_le, LE.le.trans_lt, Finset.sdiff_disjoint, Filter.mem_map, norm_nonneg, Multipliable.eventually_bounded_finsetProd, Finset.sdiff_union_distrib, Set.mem_preimage, SummationFilter.unconditional