-- Mathlib/MeasureTheory/Order/UpperLower.lean
/-- If we can fit a small ball inside a set `s` intersected with any neighborhood of `x`, then the
density of `s` near `x` is not `0`.

Along with `aux₁`, this proves that `x` is a Lebesgue point of `s`. This will be used to prove that
the frontier of an order-connected set is null. -/
private lemma aux₀
    (h : ∀ δ, 0 < δ →
      ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior s) :
    ¬Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)
        (𝓝 0)

Proof state:
case intro.intro.intro.refine_2
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 0)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
⊢ ENNReal.ofReal (4⁻¹ ^ Fintype.card ι) ≤
    ((fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) ∘ ε) n

Tactic:
calc
  ENNReal.ofReal (4⁻¹ ^ Fintype.card ι)
    = volume (closedBall (f (ε n) (hε' n)) (ε n / 4)) / volume (closedBall x (ε n)) := ?_
  _ ≤ volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n)) := by
    gcongr
    exact subset_inter ((hf₁ _ <| hε' n).trans interior_subset_closure) <| hf₀ _ <| hε' n

Result:
case intro.intro.intro.refine_2
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 0)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
⊢ ENNReal.ofReal (4⁻¹ ^ Fintype.card ι) = volume (closedBall (f (ε n) ⋯) (ε n / 4)) / volume (closedBall x (ε n))

Proof state:
case intro.intro.intro.refine_2
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 0)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ ENNReal.ofReal (4⁻¹ ^ Fintype.card ι) = volume (closedBall (f (ε n) ⋯) (ε n / 4)) / volume (closedBall x (ε n))

Tactic:
rw [Real.volume_pi_closedBall, Real.volume_pi_closedBall, ← ENNReal.ofReal_div_of_pos,
  ← div_pow, mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]

Result:
case intro.intro.intro.refine_2
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 0)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ ε n ≠ 0

case intro.intro.intro.refine_2
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 0)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ 0 < (2 * ε n) ^ Fintype.card ι

case intro.intro.intro.refine_2.hr
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 0)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ 0 ≤ ε n

case intro.intro.intro.refine_2.hr
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 0)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ 0 ≤ ε n / 4

Proof state:
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 0)
⊢ False

Tactic:
obtain ⟨ε, -, hε', hε₀⟩ := exists_seq_strictAnti_tendsto_nhdsWithin (0 : ℝ)

Result:
case intro.intro.intro
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 0)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
⊢ False