-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
left :
  ∀ (K : Finset ι) (hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2),
    Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
      (⨅ (i : ι),
        ⨅ (i_1 : ℝ≥0∞), ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1})
      {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}
left₁ :
  ∀ (K : Finset ι) (hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2) (δ : ℝ≥0∞)
    (δpos : 0 < δ) (hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2),
    Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
      (⨅ (i : ι),
        ⨅ (i_1 : ℝ≥0∞), ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1})
      {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}
left₂ :
  ∀ (K : Finset ι) (hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2) (δ : ℝ≥0∞)
    (δpos : 0 < δ) (hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2) (i : (↑(↑K : Set ι) : Type u_2)),
    Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
      (⨅ (i_1 : ℝ≥0∞),
        ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 (↑i : ι)) (a.2 (↑i : ι)) < i_1})
      {p : ((i : ι) → F i) × ((i : ι) → F i) | edist (p.1 (↑i : ι)) (p.2 (↑i : ι)) < δ}
left₃ left₄ :
  ∀ (K : Finset ι) (hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2) (δ : ℝ≥0∞)
    (δpos : 0 < δ) (hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2),
    Subset (α := Set (((i : ι) → F i) × ((i : ι) → F i)))
      (⋂ (i : (↑(↑K : Set ι) : Type u_2)),
        {p : ((i : ι) → F i) × ((i : ι) → F i) | edist (p.1 (↑i : ι)) (p.2 (↑i : ι)) < δ})
      {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}
⊢ Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (i : ι),
      ⨅ (i_1 : ℝ≥0∞), ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1})
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}

Tactic:
classical
      obtain ⟨K, hK⟩ : ∃ K : Finset ι, ∑' i : {j // j ∉ K}, 2⁻¹ ^ encode (i : ι) < ε / 2 :=
        ((tendsto_order.1 <| ENNReal.tendsto_tsum_compl_atTop_zero
          (tsum_geometric_encode_lt_top ENNReal.one_half_lt_one).ne).2 _
            <| by simpa using hε.ne').exists
      obtain ⟨δ, δpos, hδ⟩ : ∃ δ, 0 < δ ∧ δ * K.card < ε / 2 :=
        ENNReal.exists_pos_mul_lt (by simp) (by simpa using hε.ne')
      apply @mem_iInf_of_iInter _ _ _ _ _ K.finite_toSet fun i =>
          {p : (∀ i : ι, F i) × ∀ i : ι, F i | edist (p.fst i) (p.snd i) < δ}
      · rintro ⟨i, hi⟩
        refine mem_iInf_of_mem δ (mem_iInf_of_mem δpos ?_)
        simp only [mem_principal, Subset.rfl]
      · rintro ⟨x, y⟩ hxy
        simp only [mem_iInter, mem_setOf_eq, SetCoe.forall, Finset.mem_coe] at hxy
        calc
          edist x y = ∑' i : ι, min (2⁻¹ ^ encode i) (edist (x i) (y i)) := rfl
          _ = ∑ i ∈ K, min (2⁻¹ ^ encode i) (edist (x i) (y i)) +
                ∑' i : ↑(K : Set ι)ᶜ, min (2⁻¹ ^ encode (i : ι)) (edist (x i) (y i)) :=
            (ENNReal.sum_add_tsum_compl ..).symm
          _ ≤ ∑ i ∈ K, edist (x i) (y i) + ∑' i : ↑(K : Set ι)ᶜ, 2⁻¹ ^ encode (i : ι) := by
            gcongr
            · apply min_le_right
            · apply min_le_left
          _ < ∑ _i ∈ K, δ + ε / 2 := by
            refine ENNReal.add_lt_add_of_le_of_lt (by simpa using fun i hi ↦ (hxy i hi).ne_top) ?_
              hK
            gcongr with i hi
            exact (hxy i hi).le
          _ ≤ ε / 2 + ε / 2 := by gcongr; simpa [mul_comm] using hδ.le
          _ = ε := ENNReal.add_halves _

Premises used: tsum_geometric_encode_lt_top, Finset, ENNReal.exists_pos_mul_lt, ENNReal.tendsto_tsum_compl_atTop_zero, tendsto_order, Set.mem_setOf_eq, Set.mem_iInter, Filter.Eventually.exists, Filter.mem_principal, Set.Subset.rfl, And.right, ENNReal.sum_add_tsum_compl, ENNReal.add_halves, Iff.mp, ENNReal.one_half_lt_one, SetCoe.forall, Eq.symm, Finset.mem_coe, rfl, LT.lt.ne