/-- First version, with pointwise equicontinuity and range in a compact space. -/
theorem arzela_ascoli₁ [CompactSpace β] (A : Set (α →ᵇ β)) (closed : IsClosed A)
    (H : Equicontinuous ((↑) : A → α → β)) : IsCompact A := by
  simp_rw [Equicontinuous, Metric.equicontinuousAt_iff_pair] at H
  refine' isCompact_of_totallyBounded_isClosed _ closed
  refine' totallyBounded_of_finite_discretization fun ε ε0 => _
  rcases exists_between ε0 with ⟨ε₁, ε₁0, εε₁⟩
  let ε₂ := ε₁ / 2 / 2
  /- We have to find a finite discretization of `u`, i.e., finite information
    that is sufficient to reconstruct `u` up to `ε`. This information will be
    provided by the values of `u` on a sufficiently dense set `tα`,
    slightly translated to fit in a finite `ε₂`-dense set `tβ` in the image. Such
    sets exist by compactness of the source and range. Then, to check that these
    data determine the function up to `ε`, one uses the control on the modulus of
    continuity to extend the closeness on tα to closeness everywhere. -/
  have ε₂0 : ε₂ > 0 := half_pos (half_pos ε₁0)
  have : ∀ x : α, ∃ U, x ∈ U ∧ IsOpen U ∧
      ∀ y ∈ U, ∀ z ∈ U, ∀ {f : α →ᵇ β}, f ∈ A → dist (f y) (f z) < ε₂ := fun x =>
    let ⟨U, nhdsU, hU⟩ := H x _ ε₂0
    let ⟨V, VU, openV, xV⟩ := _root_.mem_nhds_iff.1 nhdsU
    ⟨V, xV, openV, fun y hy z hz f hf => hU y (VU hy) z (VU hz) ⟨f, hf⟩⟩
  choose U hU using this
  /- For all `x`, the set `hU x` is an open set containing `x` on which the elements of `A`
    fluctuate by at most `ε₂`.
    We extract finitely many of these sets that cover the whole space, by compactness. -/
  rcases isCompact_univ.elim_finite_subcover_image (fun x _ => (hU x).2.1) fun x _ =>
      mem_biUnion (mem_univ _) (hU x).1 with
    ⟨tα, _, ⟨_⟩, htα⟩
  -- `tα: Set α`, `htα : univ ⊆ ⋃x ∈ tα, U x`
  rcases @finite_cover_balls_of_compact β _ _ isCompact_univ _ ε₂0 with ⟨tβ, _, ⟨_⟩, htβ⟩
  -- `tβ : Set β`, `htβ : univ ⊆ ⋃y ∈ tβ, ball y ε₂`
  -- Associate to every point `y` in the space a nearby point `F y` in `tβ`
  choose F hF using fun y => show ∃ z ∈ tβ, dist y z < ε₂ by simpa using htβ (mem_univ y)
  -- `F : β → β`, `hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂`
  /- Associate to every function a discrete approximation, mapping each point in `tα`
    to a point in `tβ` close to its true image by the function. -/
  refine' ⟨tα → tβ, by infer_instance, fun f a => ⟨F (f.1 a), (hF (f.1 a)).1⟩, _⟩
  rintro ⟨f, hf⟩ ⟨g, hg⟩ f_eq_g
  -- If two functions have the same approximation, then they are within distance `ε`
  refine' lt_of_le_of_lt ((dist_le <| le_of_lt ε₁0).2 fun x => _) εε₁
  obtain ⟨x', x'tα, hx'⟩ := mem_iUnion₂.1 (htα (mem_univ x))
  calc
    dist (f x) (g x) ≤ dist (f x) (f x') + dist (g x) (g x') + dist (f x') (g x') :=
      dist_triangle4_right _ _ _ _
    _ ≤ ε₂ + ε₂ + ε₁ / 2 := by
      refine' le_of_lt (add_lt_add (add_lt_add _ _) _)
      · exact (hU x').2.2 _ hx' _ (hU x').1 hf
      · exact (hU x').2.2 _ hx' _ (hU x').1 hg
      · 

Proof state:
case refine'_3
F✝ : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
f✝ g✝ : α →ᵇ β
x✝ : α
C : ℝ
inst✝ : CompactSpace β
A : Set (α →ᵇ β)
closed : IsClosed A
H : ∀ (x₀ : α), ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ↑A), dist (↑i x) (↑i x') < ε
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
εε₁ : ε₁ < ε
ε₂ : ℝ := ε₁ / 2 / 2
ε₂0 : ε₂ > 0
U : α → Set α
hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ y ∈ U x, ∀ z ∈ U x, ∀ {f : α →ᵇ β}, f ∈ A → dist (f y) (f z) < ε₂
tα : Set α
left✝¹ : tα ⊆ univ
htα : univ ⊆ ⋃ i ∈ tα, U i
a✝¹ : Fintype ↑tα
tβ : Set β
left✝ : tβ ⊆ univ
htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂
a✝ : Fintype ↑tβ
F : β → β
hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂
f : α →ᵇ β
hf : f ∈ A
g : α →ᵇ β
hg : g ∈ A
f_eq_g :
  (fun f a => { val := F (↑f ↑a), property := (_ : F (↑f ↑a) ∈ tβ) }) { val := f, property := hf } =
    (fun f a => { val := F (↑f ↑a), property := (_ : F (↑f ↑a) ∈ tβ) }) { val := g, property := hg }
x x' : α
x'tα : x' ∈ tα
hx' : x ∈ U x'
⊢ dist (f x') (g x') < ε₁ / 2

Next tactic:
have F_f_g : F (f x') = F (g x') :=
          (congr_arg (fun f : tα → tβ => (f ⟨x', x'tα⟩ : β)) f_eq_g : _)