/-- Given a countable family of metric spaces, one may put a distance on their product `Π i, E i`,
defining the right topology and uniform structure. It is highly non-canonical, though, and therefore
not registered as a global instance.
The distance we use here is `dist x y = ∑' n, min (1/2)^(encode i) (dist (x n) (y n))`. -/
protected def metricSpace : MetricSpace (∀ i, F i) where
  dist_self x := by simp [dist_eq_tsum]
  dist_comm x y := by simp [dist_eq_tsum, dist_comm]
  dist_triangle x y z :=
    have I : ∀ i, min ((1 / 2) ^ encode i : ℝ) (dist (x i) (z i)) ≤
        min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i)) +
          min ((1 / 2) ^ encode i : ℝ) (dist (y i) (z i)) := fun i =>
      calc
        min ((1 / 2) ^ encode i : ℝ) (dist (x i) (z i)) ≤
            min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i) + dist (y i) (z i)) :=
          min_le_min le_rfl (dist_triangle _ _ _)
        _ = min ((1 / 2) ^ encode i : ℝ) (min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i)) +
              min ((1 / 2) ^ encode i : ℝ) (dist (y i) (z i))) := by
          convert congr_arg ((↑) : ℝ≥0 → ℝ)
            (min_add_distrib ((1 / 2 : ℝ≥0) ^ encode i) (nndist (x i) (y i))
              (nndist (y i) (z i)))
        _ ≤ min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i)) +
              min ((1 / 2) ^ encode i : ℝ) (dist (y i) (z i)) :=
          min_le_right _ _
    calc dist x z ≤ ∑' i, (min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i)) +
          min ((1 / 2) ^ encode i : ℝ) (dist (y i) (z i))) :=
        tsum_le_tsum I (dist_summable x z) ((dist_summable x y).add (dist_summable y z))
      _ = dist x y + dist y z := tsum_add (dist_summable x y) (dist_summable y z)
  edist_dist _ _ := by exact ENNReal.coe_nnreal_eq _
  eq_of_dist_eq_zero hxy := by
    ext1 n
    rw [← dist_le_zero, ← hxy]
    apply dist_le_dist_pi_of_dist_lt
    rw [hxy]
    simp
  toUniformSpace := Pi.uniformSpace _
  uniformity_dist := by
    simp only [Pi.uniformity, comap_iInf, gt_iff_lt, preimage_setOf_eq, comap_principal,
      PseudoMetricSpace.uniformity_dist]
    apply le_antisymm
    · simp only [le_iInf_iff, le_principal_iff]
      intro ε εpos
      obtain ⟨K, hK⟩ :
        ∃ K : Finset ι, (∑' i : { j // j ∉ K }, (1 / 2 : ℝ) ^ encode (i : ι)) < ε / 2 :=
        ((tendsto_order.1 (tendsto_tsum_compl_atTop_zero fun i : ι => (1 / 2 : ℝ) ^ encode i)).2 _
            (half_pos εpos)).exists
      obtain ⟨δ, δpos, hδ⟩ : ∃ δ : ℝ, 0 < δ ∧ (K.card : ℝ) * δ < ε / 2 :=
        exists_pos_mul_lt (half_pos εpos) _
      apply @mem_iInf_of_iInter _ _ _ _ _ K.finite_toSet fun i =>
          { p : (∀ i : ι, F i) × ∀ i : ι, F i | dist (p.fst i) (p.snd i) < δ }
      · rintro ⟨i, hi⟩
        refine' mem_iInf_of_mem δ (mem_iInf_of_mem δpos _)
        simp only [Prod.forall, imp_self, mem_principal, Subset.rfl]
      · rintro ⟨x, y⟩ hxy
        simp only [mem_iInter, mem_setOf_eq, SetCoe.forall, Finset.mem_range, Finset.mem_coe] at hxy
        calc
          dist x y = ∑' i : ι, min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i)) := rfl
          _ = (∑ i in K, min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i))) +
                ∑' i : ↑(K : Set ι)ᶜ, min ((1 / 2) ^ encode (i : ι) : ℝ) (dist (x i) (y i)) :=
            (sum_add_tsum_compl (dist_summable _ _)).symm
          _ ≤ (∑ i in K, dist (x i) (y i)) +
                ∑' i : ↑(K : Set ι)ᶜ, ((1 / 2) ^ encode (i : ι) : ℝ) := by
            refine' add_le_add (Finset.sum_le_sum fun i _ => min_le_right _ _) _
            refine' tsum_le_tsum (fun i => min_le_left _ _) _ _
            · apply Summable.subtype (dist_summable x y) (↑K : Set ι)ᶜ
            · apply Summable.subtype summable_geometric_two_encode (↑K : Set ι)ᶜ
          _ < (∑ _i in K, δ) + ε / 2 := by
            

Proof state:
E : ℕ → Type u_1
ι : Type u_2
inst✝¹ : Encodable ι
F : ι → Type u_3
inst✝ : (i : ι) → MetricSpace (F i)
ε : ℝ
εpos : 0 < ε
K : Finset ι
hK : ∑' (i : { j // j ∉ K }), (1 / 2) ^ encode ↑i < ε / 2
δ : ℝ
δpos : 0 < δ
hδ : ↑(Finset.card K) * δ < ε / 2
x y : (i : ι) → F i
hxy : ∀ x_1 ∈ K, dist (x x_1) (y x_1) < δ
⊢ ∑ i in K, dist (x i) (y i) + ∑' (i : ↑(↑K)ᶜ), (1 / 2) ^ encode ↑i < ∑ _i in K, δ + ε / 2

Next tactic:
apply add_lt_add_of_le_of_lt _ hK