/-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
    [CompleteSpace α] : BaireSpace α := by
  refine' ⟨fun f ho hd => _⟩
  let B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
  have Bpos : ∀ n, 0 < B n := by
    intro n
    simp only [one_div, one_mul, ENNReal.inv_pos]
    exact pow_ne_top two_ne_top
  /- Translate the density assumption into two functions `center` and `radius` associating
    to any n, x, δ, δpos a center and a positive radius such that
    `closedBall center radius` is included both in `f n` and in `closedBall x δ`.
    We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/
  have : ∀ n x δ, δ ≠ 0 → ∃ y r, 0 < r ∧ r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by
    intro n x δ δpos
    have : x ∈ closure (f n) := hd n x
    rcases EMetric.mem_closure_iff.1 this (δ / 2) (ENNReal.half_pos δpos) with ⟨y, ys, xy⟩
    rw [edist_comm] at xy
    obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n :=
      nhds_basis_closed_eball.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys)
    refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩
    show 0 < min (min (δ / 2) r) (B (n + 1))
    exact lt_min (lt_min (ENNReal.half_pos δpos) rpos) (Bpos (n + 1))
    show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1)
    exact min_le_right _ _
    show z ∈ closedBall x δ
    exact
      calc
        edist z x ≤ edist z y + edist y x := edist_triangle _ _ _
        _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz (le_of_lt xy))
        _ ≤ δ / 2 + δ / 2 := (add_le_add (le_trans (min_le_left _ _) (min_le_left _ _)) le_rfl)
        _ = δ := ENNReal.add_halves δ
    show z ∈ f n
    exact hr (calc
      edist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz
      _ ≤ r := le_trans (min_le_left _ _) (min_le_right _ _))
  choose! center radius Hpos HB Hball using this
  refine' fun x => (mem_closure_iff_nhds_basis nhds_basis_closed_eball).2 fun ε εpos => _
  /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all
    `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed
    ball `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that
    `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a
    limit which belongs to all the `f n`. -/
  let F : ℕ → α × ℝ≥0∞ := fun n =>
    Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2)
  let c : ℕ → α := fun n => (F n).1
  let r : ℕ → ℝ≥0∞ := fun n => (F n).2
  have rpos : ∀ n, 0 < r n := by
    intro n
    induction' n with n hn
    exact lt_min εpos (Bpos 0)
    exact Hpos n (c n) (r n) hn.ne'
  have r0 : ∀ n, r n ≠ 0 := fun n => (rpos n).ne'
  have rB : ∀ n, r n ≤ B n := by
    intro n
    induction' n with n _
    exact min_le_right _ _
    exact HB n (c n) (r n) (r0 n)
  have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n :=
    fun n => Hball n (c n) (r n) (r0 n)
  have cdist : ∀ n, edist (c n) (c (n + 1)) ≤ B n := by
    intro n
    rw [edist_comm]
    have A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) := mem_closedBall_self
    have I :=
      calc
        closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) :=
          Subset.trans (incl n) (inter_subset_left _ _)
        _ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n)
    exact I A
  have : CauchySeq c := cauchySeq_of_edist_le_geometric_two _ one_ne_top cdist
  -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`.
  rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩
  -- this point `y` will be the desired point. We will check that it belongs to all
  -- `f n` and to `ball x ε`.
  use y
  simp only [exists_prop, Set.mem_iInter]
  have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by
    intro n
    refine' Nat.le_induction _ fun m _ h => _
    · exact Subset.refl _
    · exact Subset.trans (incl m) (Subset.trans (inter_subset_left _ _) h)
  

Proof state:
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ : ℝ≥0∞), δ ≠ 0 → 0 < radius n x δ
HB : ∀ (n : ℕ) (x : α) (δ : ℝ≥0∞), δ ≠ 0 → radius n x δ ≤ B (n + 1)
Hball : ∀ (n : ℕ) (x : α) (δ : ℝ≥0∞), δ ≠ 0 → closedBall (center n x δ) (radius n x δ) ⊆ closedBall x δ ∩ f n
x : α
ε : ℝ≥0∞
εpos : 0 < ε
F : ℕ → α × ℝ≥0∞ := fun n => Nat.recOn n (x, min ε (B 0)) fun n p => (center n p.1 p.2, radius n p.1 p.2)
c : ℕ → α := fun n => (F n).1
r : ℕ → ℝ≥0∞ := fun n => (F n).2
rpos : ∀ (n : ℕ), 0 < r n
r0 : ∀ (n : ℕ), r n ≠ 0
rB : ∀ (n : ℕ), r n ≤ B n
incl : ∀ (n : ℕ), closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n
cdist : ∀ (n : ℕ), edist (c n) (c (n + 1)) ≤ B n
this : CauchySeq c
y : α
ylim : Tendsto c atTop (𝓝 y)
I : ∀ (n m : ℕ), m ≥ n → closedBall (c m) (r m) ⊆ closedBall (c n) (r n)
⊢ (∀ (i : ℕ), y ∈ f i) ∧ y ∈ closedBall x ε

Next tactic:
have yball : ∀ n, y ∈ closedBall (c n) (r n) := by
    intro n
    refine' isClosed_ball.mem_of_tendsto ylim _
    refine' (Filter.eventually_ge_atTop n).mono fun m hm => _
    exact I n m hm mem_closedBall_self