Module: Mathlib.Topology.MetricSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : EMetricSpace α
inst✝¹ : EMetricSpace β
s : Set α
inst✝ : CompleteSpace α
⊢ CompleteSpace (Closeds α)

Tactic:
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 [B, ENNReal.pow_pos]
  have B_ne_top : ∀ n, B n ≠ ⊤ := by finiteness
  /- 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 EMetric.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 (s := s (n + l)) 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 => (this l z).choose
      simp only [Nat.add_zero, Nat.rec_zero, true_and]
      exact fun k => (this k _).choose_spec
    -- it follows from the previous bound that `z` is a Cauchy sequence
    have : CauchySeq fun k => (z k : α) := cauchySeq_of_edist_le_geometric_two (B n) (B_ne_top n) hz
    -- therefore, it converges
    rcases cauchySeq_tendsto_of_complete this with ⟨y, y_lim⟩
    use y
    -- the limit point `y` will be the desired point, in `t0` and close to our initial point `x`.
    -- First, we check it belongs to `t0`.
    have : y ∈ t0 :=
      mem_iInter.2 fun k =>
        mem_closure_of_tendsto y_lim
          (by
            simp only [exists_prop, Set.mem_iUnion, Filter.eventually_atTop]
            exact ⟨k, fun m hm => ⟨n + m, by lia, (z m).2⟩⟩)
    use this
    -- Then, we check that `y` is close to `x = z n`. This follows from the fact that `y`
    -- is the limit of `z k`, and the distance between `z n` and `z k` has already been estimated.
    rw [← hz₀]
    exact edist_le_of_edist_le_geometric_two_of_tendsto₀ (B n) hz y_lim
  have I2 : ∀ n, ∀ x ∈ t0, ∃ y ∈ s n, edist x y ≤ 2 * B n := by
    /- For the (much easier) reverse inequality, we start from a point `x ∈ t0` and we want
            to find a point `y ∈ s n` which is close to `x`.
            `x` belongs to `t0`, the intersection of the closures. In particular, it is well
            approximated by a point `z` in `⋃m≥n, s m`, say in `s m`. Since `s m` and
            `s n` are close, this point is itself well approximated by a point `y` in `s n`,
            as required. -/
    intro n x xt0
    have : x ∈ closure (⋃ m ≥ n, s m : Set α) := by apply mem_iInter.1 xt0 n
    obtain ⟨z : α, hz, Dxz : edist x z < B n⟩ := EMetric.mem_closure_iff.1 this (B n) (B_pos n)
    simp only [exists_prop, Set.mem_iUnion] at hz
    obtain ⟨m : ℕ, m_ge_n : m ≥ n, hm : z ∈ (s m : Set α)⟩ := hz
    have : hausdorffEDist (s m : Set α) (s n) < B n := hs n m n m_ge_n (le_refl n)
    obtain ⟨y : α, hy : y ∈ (s n : Set α), Dzy : edist z y < B n⟩ :=
      exists_edist_lt_of_hausdorffEDist_lt hm this
    exact
      ⟨y, hy,
        calc
          edist x y ≤ edist x z + edist z y := edist_triangle _ _ _
          _ ≤ B n + B n := by gcongr
          _ = 2 * B n := (two_mul _).symm
          ⟩
  -- Deduce from the above inequalities that the distance between `s n` and `t0` is at most `2 B n`.
  have main : ∀ n : ℕ, edist (s n) t ≤ 2 * B n := fun n =>
    hausdorffEDist_le_of_mem_edist (I1 n) (I2 n)
  -- from this, the convergence of `s n` to `t0` follows.
  refine EMetric.tendsto_atTop.2 fun ε εpos => ?_
  have : Tendsto (fun n => 2 * B n) atTop (𝓝 (2 * 0)) :=
    ENNReal.Tendsto.const_mul (ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one <|
      by simp) (Or.inr <| by simp)
  rw [mul_zero] at this
  obtain ⟨N, hN⟩ : ∃ N, ∀ b ≥ N, ε > 2 * B b :=
    ((tendsto_order.1 this).2 ε εpos).exists_forall_of_atTop
  exact ⟨N, fun n hn => lt_of_le_of_lt (main n) (hN n hn)⟩

Goal state after: