/-- 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)
    refine'
      ⟨y⁻¹ * g, by
        simpa only [div_eq_mul_inv, mul_inv_rev, inv_inv, mul_inv_cancel_left] using y_mem, _⟩
    rw [QuotientGroup.mk_mul, QuotientGroup.mk_inv, hy, hg, inv_div, div_mul_cancel']
  /- Inductively construct a subsequence `φ : ℕ → ℕ` using `key₀` so that if `a b : ℕ` exceed
    `φ (n + 1)`, then we may find lifts whose quotients lie within `u n`. -/
  set φ : ℕ → ℕ := fun n => Nat.recOn n (choose <| key₀ 0 0) fun k yk => choose <| key₀ (k + 1) yk
  have hφ :
    ∀ n : ℕ,
      φ n < φ (n + 1) ∧
        ∀ a b : ℕ,
          φ (n + 1) ≤ a →
            φ (n + 1) ≤ b → ∀ g : G, x b = g → ∃ g' : G, g / g' ∈ u (n + 1) ∧ x a = g' :=
    fun n => choose_spec (key₀ (n + 1) (φ n))
  /- Inductively construct a sequence `x' n : G` of lifts of `x (φ (n + 1))` such that quotients of
    successive terms lie in `x' n / x' (n + 1) ∈ u (n + 1)`. We actually need the proofs that each
    term is a lift to construct the next term, so we use a Σ-type. -/
  set x' : ∀ n, PSigma fun g : G => x (φ (n + 1)) = g := fun n =>
    Nat.recOn n
      ⟨choose (QuotientGroup.mk_surjective (x (φ 1))),
        (choose_spec (QuotientGroup.mk_surjective (x (φ 1)))).symm⟩
      fun k hk =>
      ⟨choose <| (hφ k).2 _ _ (hφ (k + 1)).1.le le_rfl hk.fst hk.snd,
        (choose_spec <| (hφ k).2 _ _ (hφ (k + 1)).1.le le_rfl hk.fst hk.snd).2⟩
  have hx' : ∀ n : ℕ, (x' n).fst / (x' (n + 1)).fst ∈ u (n + 1) := fun n =>
    (choose_spec <| (hφ n).2 _ _ (hφ (n + 1)).1.le le_rfl (x' n).fst (x' n).snd).1
  /- The sequence `x'` is Cauchy. This is where we exploit the condition on `u`. The key idea
    is to show by decreasing induction that `x' m / x' n ∈ u m` if `m ≤ n`. -/
  have x'_cauchy : CauchySeq fun n => (x' n).fst := by
    have h𝓤G : (𝓤 G).HasBasis (fun _ => True) fun i => { x | x.snd / x.fst ∈ u i } := by
      simpa [uniformity_eq_comap_nhds_one'] using hu.toHasBasis.comap _
    

Proof state:
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
hx : CauchySeq x
key₀ : ∀ (i j : ℕ), ∃ M, j < M ∧ ∀ (a b : ℕ), M ≤ a → M ≤ b → ∀ (g : G), x b = ↑g → ∃ g', g / g' ∈ u i ∧ x a = ↑g'
φ : ℕ → ℕ :=
  fun n =>
    Nat.recOn n
      (choose (_ : ∃ M, 0 < M ∧ ∀ (a b : ℕ), M ≤ a → M ≤ b → ∀ (g : G), x b = ↑g → ∃ g', g / g' ∈ u 0 ∧ x a = ↑g'))
      fun k yk =>
      choose (_ : ∃ M, yk < M ∧ ∀ (a b : ℕ), M ≤ a → M ≤ b → ∀ (g : G), x b = ↑g → ∃ g', g / g' ∈ u (k + 1) ∧ x a = ↑g')
hφ :
  ∀ (n : ℕ),
    φ n < φ (n + 1) ∧
      ∀ (a b : ℕ), φ (n + 1) ≤ a → φ (n + 1) ≤ b → ∀ (g : G), x b = ↑g → ∃ g', g / g' ∈ u (n + 1) ∧ x a = ↑g'
x' : (n : ℕ) → (g : G) ×' x (φ (n + 1)) = ↑g :=
  fun n =>
    Nat.recOn n { fst := choose (_ : ∃ a, ↑a = x (φ 1)), snd := (_ : x (φ 1) = ↑(choose (_ : ∃ a, ↑a = x (φ 1)))) }
      fun k hk =>
      { fst := choose (_ : ∃ g', hk.fst / g' ∈ u (k + 1) ∧ x (φ (k + 1 + 1)) = ↑g'),
        snd := (_ : x (φ (k + 1 + 1)) = ↑(choose (_ : ∃ g', hk.fst / g' ∈ u (k + 1) ∧ x (φ (k + 1 + 1)) = ↑g'))) }
hx' : ∀ (n : ℕ), (x' n).fst / (x' (n + 1)).fst ∈ u (n + 1)
h𝓤G : HasBasis (𝓤 G) (fun x => True) fun i => {x | x.2 / x.1 ∈ u i}
⊢ CauchySeq fun n => (x' n).fst

Next tactic:
rw [h𝓤G.cauchySeq_iff']