/-- In a complete space, the type of closed subsets is complete for the
Hausdorff edistance. -/
instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) := by
  /- We will show that, if a sequence of sets `s n` satisfies
    `edist (s n) (s (n+1)) < 2^{-n}`, then it converges. This is enough to guarantee
    completeness, by a standard completeness criterion.
    We use the shorthand `B n = 2^{-n}` in ennreal. -/
  let B : ℕ → ℝ≥0∞ := fun n => 2⁻¹ ^ n
  have B_pos : ∀ n, (0 : ℝ≥0∞) < B n := by simp [ENNReal.pow_pos]
  have B_ne_top : ∀ n, B n ≠ ⊤ := by simp [ENNReal.pow_ne_top]
  /- Consider a sequence of closed sets `s n` with `edist (s n) (s (n+1)) < B n`.
    We will show that it converges. The limit set is `t0 = ⋂n, closure (⋃m≥n, s m)`.
    We will have to show that a point in `s n` is close to a point in `t0`, and a point
    in `t0` is close to a point in `s n`. The completeness then follows from a
    standard criterion. -/
  refine' complete_of_convergent_controlled_sequences B B_pos fun s hs => _
  let t0 := ⋂ n, closure (⋃ m ≥ n, s m : Set α)
  let t : Closeds α := ⟨t0, isClosed_iInter fun _ => isClosed_closure⟩
  use t
  -- The inequality is written this way to agree with `edist_le_of_edist_le_geometric_of_tendsto₀`
  have I1 : ∀ n, ∀ x ∈ s n, ∃ y ∈ t0, edist x y ≤ 2 * B n := by
    /- This is the main difficulty of the proof. Starting from `x ∈ s n`, we want
           to find a point in `t0` which is close to `x`. Define inductively a sequence of
           points `z m` with `z n = x` and `z m ∈ s m` and `edist (z m) (z (m+1)) ≤ B m`. This is
           possible since the Hausdorff distance between `s m` and `s (m+1)` is at most `B m`.
           This sequence is a Cauchy sequence, therefore converging as the space is complete, to
           a limit which satisfies the required properties. -/
    intro n x hx
    obtain ⟨z, hz₀, hz⟩ :
      ∃ z : ∀ l, s (n + l), (z 0 : α) = x ∧ ∀ k, edist (z k : α) (z (k + 1) : α) ≤ B n / 2 ^ k := by
      -- We prove existence of the sequence by induction.
      have : ∀ (l) (z : s (n + l)), ∃ z' : s (n + l + 1), edist (z : α) z' ≤ B n / 2 ^ l := by
        intro l z
        obtain ⟨z', z'_mem, hz'⟩ : ∃ z' ∈ s (n + l + 1), edist (z : α) z' < B n / 2 ^ l := by
          refine' exists_edist_lt_of_hausdorffEdist_lt _ _
          · exact (s (n + l) : Set α)
          · exact z.2
          simp only [ENNReal.inv_pow, div_eq_mul_inv]
          rw [← pow_add]
          apply hs <;> simp
        exact ⟨⟨z', z'_mem⟩, le_of_lt hz'⟩
      use fun k => Nat.recOn k ⟨x, hx⟩ fun l z => choose (this l z)
      simp only [Nat.add_zero, Nat.zero_eq, Nat.rec_zero, Nat.rec_add_one, true_and]
      exact fun k => choose_spec (this k _)
    -- it follows from the previous bound that `z` is a Cauchy sequence
    

Proof state:
case intro.intro
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun n => 2⁻¹ ^ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ⊤
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α := ⋂ n, closure (⋃ m, ⋃ (_ : m ≥ n), ↑(s m))
t : Closeds α := { carrier := t0, closed' := (_ : IsClosed (⋂ i, closure (⋃ m, ⋃ (_ : m ≥ i), ↑(s m)))) }
n : ℕ
x : α
hx : x ∈ s n
z : (l : ℕ) → ↥(s (n + l))
hz₀ : ↑(z 0) = x
hz : ∀ (k : ℕ), edist ↑(z k) ↑(z (k + 1)) ≤ B n / 2 ^ k
⊢ ∃ y ∈ t0, edist x y ≤ 2 * B n

Next tactic:
have : CauchySeq fun k => (z k : α) := cauchySeq_of_edist_le_geometric_two (B n) (B_ne_top n) hz