/-- The quotient `G ⧸ N` of a complete first countable topological group `G` by a normal subgroup
is itself complete. [N. Bourbaki, *General Topology*, IX.3.1 Proposition 4][bourbaki1966b]

Because a topological group is not equipped with a `UniformSpace` instance by default, we must
explicitly provide it in order to consider completeness. See `QuotientGroup.completeSpace` for a
version in which `G` is already equipped with a uniform structure. -/
@[to_additive "The quotient `G ⧸ N` of a complete first countable topological additive group
`G` by a normal additive subgroup is itself complete. Consequently, quotients of Banach spaces by
subspaces are complete. [N. Bourbaki, *General Topology*, IX.3.1 Proposition 4][bourbaki1966b]

Because an additive topological group is not equipped with a `UniformSpace` instance by default,
we must explicitly provide it in order to consider completeness. See
`QuotientAddGroup.completeSpace` for a version in which `G` is already equipped with a uniform
structure."]
instance QuotientGroup.completeSpace' (G : Type u) [Group G] [TopologicalSpace G]
    [TopologicalGroup G] [FirstCountableTopology G] (N : Subgroup G) [N.Normal]
    [@CompleteSpace G (TopologicalGroup.toUniformSpace G)] :
    @CompleteSpace (G ⧸ N) (TopologicalGroup.toUniformSpace (G ⧸ N)) := by
  /- Since `G ⧸ N` is a topological group it is a uniform space, and since `G` is first countable
    the uniformities of both `G` and `G ⧸ N` are countably generated. Moreover, we may choose a
    sequential antitone neighborhood basis `u` for `𝓝 (1 : G)` so that `(u (n + 1)) ^ 2 ⊆ u n`, and
    this descends to an antitone neighborhood basis `v` for `𝓝 (1 : G ⧸ N)`. Since `𝓤 (G ⧸ N)` is
    countably generated, it suffices to show any Cauchy sequence `x` converges. -/
  letI : UniformSpace (G ⧸ N) := TopologicalGroup.toUniformSpace (G ⧸ N)
  letI : UniformSpace G := TopologicalGroup.toUniformSpace G
  haveI : (𝓤 (G ⧸ N)).IsCountablyGenerated := comap.isCountablyGenerated _ _
  obtain ⟨u, hu, u_mul⟩ := TopologicalGroup.exists_antitone_basis_nhds_one G
  obtain ⟨hv, v_anti⟩ := @HasAntitoneBasis.map _ _ _ _ _ _ ((↑) : G → G ⧸ N) hu
  rw [← QuotientGroup.nhds_eq N 1, QuotientGroup.mk_one] at hv
  refine' UniformSpace.complete_of_cauchySeq_tendsto fun x hx => _
  /- Given `n : ℕ`, for sufficiently large `a b : ℕ`, given any lift of `x b`, we can find a lift
    of `x a` such that the quotient of the lifts lies in `u n`. -/
  have key₀ : ∀ i j : ℕ, ∃ M : ℕ, j < M ∧ ∀ a b : ℕ, M ≤ a → M ≤ b →
      ∀ g : G, x b = g → ∃ g' : G, g / g' ∈ u i ∧ x a = g' := by
    have h𝓤GN : (𝓤 (G ⧸ N)).HasBasis (fun _ => True) fun i => { x | x.snd / x.fst ∈ (↑) '' u i } :=
      by simpa [uniformity_eq_comap_nhds_one'] using hv.comap _
    rw [h𝓤GN.cauchySeq_iff] at hx
    simp only [ge_iff_le, mem_setOf_eq, forall_true_left, mem_image] at hx
    intro i j
    rcases hx i with ⟨M, hM⟩
    refine' ⟨max j M + 1, (le_max_left _ _).trans_lt (lt_add_one _), fun a b ha hb g hg => _⟩
    obtain ⟨y, y_mem, hy⟩ :=
      hM a (((le_max_right j _).trans (lt_add_one _).le).trans ha) b
        (((le_max_right j _).trans (lt_add_one _).le).trans hb)
    

Proof state:
case intro.intro.intro
G : Type u
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalGroup G
inst✝² : FirstCountableTopology G
N : Subgroup G
inst✝¹ : Subgroup.Normal N
inst✝ : CompleteSpace G
this✝¹ : UniformSpace (G ⧸ N) := TopologicalGroup.toUniformSpace (G ⧸ N)
this✝ : UniformSpace G := TopologicalGroup.toUniformSpace G
this : IsCountablyGenerated (𝓤 (G ⧸ N))
u : ℕ → Set G
hu : HasAntitoneBasis (𝓝 1) u
u_mul : ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n
hv : HasBasis (𝓝 1) (fun x => True) fun n => mk '' u n
v_anti : Antitone fun n => mk '' u n
x : ℕ → G ⧸ N
h𝓤GN : HasBasis (𝓤 (G ⧸ N)) (fun x => True) fun i => {x | x.2 / x.1 ∈ mk '' u i}
hx : ∀ (i : ℕ), ∃ N_1, ∀ (m : ℕ), N_1 ≤ m → ∀ (n : ℕ), N_1 ≤ n → ∃ x_1 ∈ u i, ↑x_1 = x n / x m
i j M : ℕ
hM : ∀ (m : ℕ), M ≤ m → ∀ (n : ℕ), M ≤ n → ∃ x_1 ∈ u i, ↑x_1 = x n / x m
a b : ℕ
ha : max j M + 1 ≤ a
hb : max j M + 1 ≤ b
g : G
hg : x b = ↑g
y : G
y_mem : y ∈ u i
hy : ↑y = x b / x a
⊢ ∃ g', g / g' ∈ u i ∧ x a = ↑g'

Next tactic:
refine'
      ⟨y⁻¹ * g, by
        simpa only [div_eq_mul_inv, mul_inv_rev, inv_inv, mul_inv_cancel_left] using y_mem, _⟩