Theorem: _private.Mathlib.MeasureTheory.Order.UpperLower.0.aux₁
File: Mathlib/MeasureTheory/Order/UpperLower.lean

Proof steps:
state_before: ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : ℝ
h : ∀ (δ : ℝ), 0 < δ → ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior sᶜ
⊢ ¬Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
state_after: ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
⊢ ¬Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
tactic: choose f hf₀ hf₁ using h
state_before: ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
⊢ ¬Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
state_after: ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
⊢ False
tactic: intro H
state_before: ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
⊢ False
state_after: case intro.intro.intro
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
⊢ False
tactic: obtain ⟨ε, -, hε', hε₀⟩ := exists_seq_strictAnti_tendsto_nhdsWithin (0 : ℝ)
state_before: case intro.intro.intro
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
⊢ False
state_after: case intro.intro.intro.refine_1
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
⊢ ENNReal.ofReal (4⁻¹ ^ Fintype.card ι) ≠ 0

case intro.intro.intro.refine_2
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
⊢ ((fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) ∘ ε) n ≤
    1 - ENNReal.ofReal (4⁻¹ ^ Fintype.card ι)
tactic: refine not_eventually.2
      (frequently_of_forall fun _ ↦ lt_irrefl $ 1 - ENNReal.ofReal (4⁻¹ ^ Fintype.card ι))
      ((Filter.Tendsto.eventually_lt tendsto_const_nhds (H.comp hε₀) $
            ENNReal.sub_lt_self ENNReal.one_ne_top one_ne_zero ?_).mono
        fun n ↦ lt_of_le_of_lt' ?_)
state_before: case intro.intro.intro.refine_1
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
⊢ ENNReal.ofReal (4⁻¹ ^ Fintype.card ι) ≠ 0

case intro.intro.intro.refine_2
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
⊢ ((fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) ∘ ε) n ≤
    1 - ENNReal.ofReal (4⁻¹ ^ Fintype.card ι)
state_after: case intro.intro.intro.refine_1
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
⊢ ENNReal.ofReal (4⁻¹ ^ Fintype.card ι) ≠ 0

case intro.intro.intro.refine_2
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
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 y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
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

ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ ENNReal.ofReal ((2 * ε n) ^ Fintype.card ι) ≠ 0

ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ 0 < ENNReal.ofReal ((2 * (ε n / 4)) ^ Fintype.card ι) →
    ENNReal.ofReal ((2 * (ε n / 4)) ^ Fintype.card ι) < ENNReal.ofReal ((2 * ε n) ^ Fintype.card ι) →
      ENNReal.ofReal ((2 * ε n) ^ Fintype.card ι) ≠ 0

case intro.intro.intro.refine_2.hr
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
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

case intro.intro.intro.refine_2.hr
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ 0 ≤ ε n

ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ MeasurableSet (closedBall (f (ε n) ⋯) (ε n / 4))

ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
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
tactic: on_goal 2 =>
    calc
      volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n))
        ≤ volume (closedBall x (ε n) \ closedBall (f (ε n) $ hε' n) (ε n / 4)) /
          volume (closedBall x (ε n)) := by
        gcongr
        rw [diff_eq_compl_inter]
        refine inter_subset_inter_left _ ?_
        rw [subset_compl_comm, ← interior_compl]
        exact hf₁ _ _
      _ = 1 - ENNReal.ofReal (4⁻¹ ^ Fintype.card ι) := ?_
    dsimp only
    have := hε' n
    rw [measure_diff (hf₀ _ _) _ ((Real.volume_pi_closedBall _ _).trans_ne ENNReal.ofReal_ne_top),
      Real.volume_pi_closedBall, Real.volume_pi_closedBall, ENNReal.sub_div fun _ _ ↦ _,
      ENNReal.div_self _ ENNReal.ofReal_ne_top, ← ENNReal.ofReal_div_of_pos, ← div_pow,
      mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]
state_before: case intro.intro.intro.refine_1
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
⊢ ENNReal.ofReal (4⁻¹ ^ Fintype.card ι) ≠ 0

case intro.intro.intro.refine_2
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
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 y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
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

ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ ENNReal.ofReal ((2 * ε n) ^ Fintype.card ι) ≠ 0

ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ 0 < ENNReal.ofReal ((2 * (ε n / 4)) ^ Fintype.card ι) →
    ENNReal.ofReal ((2 * (ε n / 4)) ^ Fintype.card ι) < ENNReal.ofReal ((2 * ε n) ^ Fintype.card ι) →
      ENNReal.ofReal ((2 * ε n) ^ Fintype.card ι) ≠ 0

case intro.intro.intro.refine_2.hr
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
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

case intro.intro.intro.refine_2.hr
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ 0 ≤ ε n

ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ MeasurableSet (closedBall (f (ε n) ⋯) (ε n / 4))

ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
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
state_after: ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ 0 < ENNReal.ofReal ((2 * (ε n / 4)) ^ Fintype.card ι) →
    ENNReal.ofReal ((2 * (ε n / 4)) ^ Fintype.card ι) < ENNReal.ofReal ((2 * ε n) ^ Fintype.card ι) →
      ENNReal.ofReal ((2 * ε n) ^ Fintype.card ι) ≠ 0

ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x y : ι → ℝ
δ : 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 sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ MeasurableSet (closedBall (f (ε n) ⋯) (ε n / 4))
tactic: all_goals try positivity