/-- For a regular topological space with second countable topology,
there exists an inducing map to `l^∞ = ℕ →ᵇ ℝ`. -/
theorem exists_inducing_l_infty : ∃ f : X → ℕ →ᵇ ℝ, Inducing f := by
  -- Choose a countable basis, and consider the set `s` of pairs of set `(U, V)` such that `U ∈ B`,
  -- `V ∈ B`, and `closure U ⊆ V`.
  rcases exists_countable_basis X with ⟨B, hBc, -, hB⟩
  let s : Set (Set X × Set X) := { UV ∈ B ×ˢ B | closure UV.1 ⊆ UV.2 }
  -- `s` is a countable set.
  haveI : Encodable s := ((hBc.prod hBc).mono (inter_subset_left _ _)).toEncodable
  -- We don't have the space of bounded (possibly discontinuous) functions, so we equip `s`
  -- with the discrete topology and deal with `s →ᵇ ℝ` instead.
  letI : TopologicalSpace s := ⊥
  haveI : DiscreteTopology s := ⟨rfl⟩
  rsuffices ⟨f, hf⟩ : ∃ f : X → s →ᵇ ℝ, Inducing f
  · exact ⟨fun x => (f x).extend (Encodable.encode' s) 0,
      (BoundedContinuousFunction.isometry_extend (Encodable.encode' s)
        (0 : ℕ →ᵇ ℝ)).embedding.toInducing.comp hf⟩
  have hd : ∀ UV : s, Disjoint (closure UV.1.1) UV.1.2ᶜ :=
    fun UV => disjoint_compl_right.mono_right (compl_subset_compl.2 UV.2.2)
  -- Choose a sequence of `εₙ > 0`, `n : s`, that is bounded above by `1` and tends to zero
  -- along the `cofinite` filter.
  obtain ⟨ε, ε01, hε⟩ : ∃ ε : s → ℝ, (∀ UV, ε UV ∈ Ioc (0 : ℝ) 1) ∧ Tendsto ε cofinite (𝓝 0) := by
    rcases posSumOfEncodable zero_lt_one s with ⟨ε, ε0, c, hεc, hc1⟩
    refine' ⟨ε, fun UV => ⟨ε0 UV, _⟩, hεc.summable.tendsto_cofinite_zero⟩
    exact (le_hasSum hεc UV fun _ _ => (ε0 _).le).trans hc1
  /- For each `UV = (U, V) ∈ s` we use Urysohn's lemma to choose a function `f UV` that is equal to
    zero on `U` and is equal to `ε UV` on the complement to `V`. -/
  have : ∀ UV : s, ∃ f : C(X, ℝ),
      EqOn f 0 UV.1.1 ∧ EqOn f (fun _ => ε UV) UV.1.2ᶜ ∧ ∀ x, f x ∈ Icc 0 (ε UV) := by
    intro UV
    rcases exists_continuous_zero_one_of_isClosed isClosed_closure
        (hB.isOpen UV.2.1.2).isClosed_compl (hd UV) with
      ⟨f, hf₀, hf₁, hf01⟩
    exact ⟨ε UV • f, fun x hx => by simp [hf₀ (subset_closure hx)], fun x hx => by simp [hf₁ hx],
      fun x => ⟨mul_nonneg (ε01 _).1.le (hf01 _).1, mul_le_of_le_one_right (ε01 _).1.le (hf01 _).2⟩⟩
  choose f hf0 hfε hf0ε using this
  have hf01 : ∀ UV x, f UV x ∈ Icc (0 : ℝ) 1 :=
    fun UV x => Icc_subset_Icc_right (ε01 _).2 (hf0ε _ _)
  -- The embedding is given by `F x UV = f UV x`.
  set F : X → s →ᵇ ℝ := fun x =>
    ⟨⟨fun UV => f UV x, continuous_of_discreteTopology⟩, 1,
      fun UV₁ UV₂ => Real.dist_le_of_mem_Icc_01 (hf01 _ _) (hf01 _ _)⟩
  have hF : ∀ x UV, F x UV = f UV x := fun _ _ => rfl
  refine' ⟨F, inducing_iff_nhds.2 fun x => le_antisymm _ _⟩
  · /- First we prove that `F` is continuous. Given `δ > 0`, consider the set `T` of `(U, V) ∈ s`
    such that `ε (U, V) ≥ δ`. Since `ε` tends to zero, `T` is finite. Since each `f` is continuous,
    we can choose a neighborhood such that `dist (F y (U, V)) (F x (U, V)) ≤ δ` for any
    `(U, V) ∈ T`. For `(U, V) ∉ T`, the same inequality is true because both `F y (U, V)` and
    `F x (U, V)` belong to the interval `[0, ε (U, V)]`. -/
    refine' (nhds_basis_closedBall.comap _).ge_iff.2 fun δ δ0 => _
    have h_fin : { UV : s | δ ≤ ε UV }.Finite := by simpa only [← not_lt] using hε (gt_mem_nhds δ0)
    have : ∀ᶠ y in 𝓝 x, ∀ UV, δ ≤ ε UV → dist (F y UV) (F x UV) ≤ δ := by
      refine' (eventually_all_finite h_fin).2 fun UV _ => _
      exact (f UV).continuous.tendsto x (closedBall_mem_nhds _ δ0)
    refine' this.mono fun y hy => (BoundedContinuousFunction.dist_le δ0.le).2 fun UV => _
    rcases le_total δ (ε UV) with hle | hle
    exacts [hy _ hle, (Real.dist_le_of_mem_Icc (hf0ε _ _) (hf0ε _ _)).trans (by rwa [sub_zero])]
  · /- Finally, we prove that each neighborhood `V` of `x : X`
    includes a preimage of a neighborhood of `F x` under `F`.
    Without loss of generality, `V` belongs to `B`.
    Choose `U ∈ B` such that `x ∈ V` and `closure V ⊆ U`.
    Then the preimage of the `(ε (U, V))`-neighborhood of `F x` is included by `V`. -/
    refine' ((nhds_basis_ball.comap _).le_basis_iff hB.nhds_hasBasis).2 _
    rintro V ⟨hVB, hxV⟩
    rcases hB.exists_closure_subset (hB.mem_nhds hVB hxV) with ⟨U, hUB, hxU, hUV⟩
    set UV : ↥s := ⟨(U, V), ⟨hUB, hVB⟩, hUV⟩
    refine' ⟨ε UV, (ε01 UV).1, fun y (hy : dist (F y) (F x) < ε UV) => _⟩
    replace hy : dist (F y UV) (F x UV) < ε UV
    exact (BoundedContinuousFunction.dist_coe_le_dist _).trans_lt hy
    contrapose! hy
    

Proof state:
case intro.intro.refine'_2.intro.intro.intro.intro
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : SecondCountableTopology X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.1 ⊆ UV.2}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).1) (↑UV).2ᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (⇑(f UV)) 0 (↑UV).1
hfε : ∀ (UV : ↑s), EqOn (⇑(f UV)) (fun x => ε UV) (↑UV).2ᶜ
hf0ε : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => (f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => (f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => (f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), (F x) UV = (f UV) x
x : X
V : Set X
hVB : V ∈ B
hxV : x ∈ V
U : Set X
hUB : U ∈ B
hxU : x ∈ U
hUV : closure U ⊆ V
UV : ↑s := { val := (U, V), property := (_ : (U, V) ∈ B ×ˢ B ∧ closure (U, V).1 ⊆ (U, V).2) }
y : X
hy : y ∉ V
⊢ ε UV ≤ dist ((F y) UV) ((F x) UV)

Next tactic:
rw [hF, hF, hfε UV hy, hf0 UV hxU, Pi.zero_apply, dist_zero_right]