-- Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
theorem tendsto_approxOn_Lp_eLpNorm [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f)
    {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (hp_ne_top : p ≠ ∞) {μ : Measure β}
    (hμ : ∀ᵐ x ∂μ, f x ∈ closure s) (hi : eLpNorm (fun x => f x - y₀) p μ < ∞) :
    Tendsto (fun n => eLpNorm (⇑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0)

Proof state:
case neg
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ⊤
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s
hi : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal
h_bound : ∀ (n : ℕ), (fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal) ≤ᶠ[ae μ] fun x => ‖f x - y₀‖ₑ ^ p.toReal
⊢ Tendsto (fun n => ∫⁻ (x : β), ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal ∂μ) atTop (𝓝 0)

Tactic:
have h_fin : (∫⁻ a : β, ‖f a - y₀‖ₑ ^ p.toReal ∂μ) ≠ ⊤ :=
  (lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top hp_zero hp_ne_top hi).ne

Result:
case neg
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ⊤
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s
hi : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal
h_bound : ∀ (n : ℕ), (fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal) ≤ᶠ[ae μ] fun x => ‖f x - y₀‖ₑ ^ p.toReal
h_fin : ∫⁻ (a : β), ‖f a - y₀‖ₑ ^ p.toReal ∂μ ≠ ⊤
⊢ Tendsto (fun n => ∫⁻ (x : β), ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal ∂μ) atTop (𝓝 0)

Proof state:
case pos
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ⊤
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s
hi : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : p = 0
⊢ Tendsto (fun n => eLpNorm (⇑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0)

Tactic:
simpa only [hp_zero, eLpNorm_exponent_zero] using tendsto_const_nhds

Result:
no goals

Proof state:
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ⊤
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s
hi : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal
h_bound : ∀ (n : ℕ), (fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal) ≤ᶠ[ae μ] fun x => ‖f x - y₀‖ₑ ^ p.toReal
h_fin : ∫⁻ (a : β), ‖f a - y₀‖ₑ ^ p.toReal ∂μ ≠ ⊤
⊢ ∀ᵐ (a : β) ∂μ, Tendsto (fun x => ‖(approxOn f hf s y₀ h₀ x) a - f a‖ₑ ^ p.toReal) atTop (𝓝 0)

Tactic:
filter_upwards [hμ] with a ha

Result:
case h
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ⊤
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s
hi : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal
h_bound : ∀ (n : ℕ), (fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal) ≤ᶠ[ae μ] fun x => ‖f x - y₀‖ₑ ^ p.toReal
h_fin : ∫⁻ (a : β), ‖f a - y₀‖ₑ ^ p.toReal ∂μ ≠ ⊤
a : β
ha : f a ∈ closure s
⊢ Tendsto (fun x => ‖(approxOn f hf s y₀ h₀ x) a - f a‖ₑ ^ p.toReal) atTop (𝓝 0)

Proof state:
case h
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ⊤
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s
hi : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal
h_bound : ∀ (n : ℕ), (fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal) ≤ᶠ[ae μ] fun x => ‖f x - y₀‖ₑ ^ p.toReal
h_fin : ∫⁻ (a : β), ‖f a - y₀‖ₑ ^ p.toReal ∂μ ≠ ⊤
a : β
ha : f a ∈ closure s
⊢ Tendsto (fun x => ‖(approxOn f hf s y₀ h₀ x) a - f a‖ₑ ^ p.toReal) atTop (𝓝 0)

Tactic:
have : Tendsto (fun n => (approxOn f hf s y₀ h₀ n) a - f a) atTop (𝓝 (f a - f a)) :=
  (tendsto_approxOn hf h₀ ha).sub tendsto_const_nhds

Result:
case h
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ⊤
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s
hi : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal
h_bound : ∀ (n : ℕ), (fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal) ≤ᶠ[ae μ] fun x => ‖f x - y₀‖ₑ ^ p.toReal
h_fin : ∫⁻ (a : β), ‖f a - y₀‖ₑ ^ p.toReal ∂μ ≠ ⊤
a : β
ha : f a ∈ closure s
this : Tendsto (fun n => (approxOn f hf s y₀ h₀ n) a - f a) atTop (𝓝 (f a - f a))
⊢ Tendsto (fun x => ‖(approxOn f hf s y₀ h₀ x) a - f a‖ₑ ^ p.toReal) atTop (𝓝 0)

Proof state:
case neg
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ⊤
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s
hi : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
⊢ Tendsto (fun n => eLpNorm (⇑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0)

Tactic:
suffices Tendsto (fun n => ∫⁻ x, ‖approxOn f hf s y₀ h₀ n x - f x‖ₑ ^ p.toReal ∂μ) atTop (𝓝 0) by
  simp only [eLpNorm_eq_lintegral_rpow_enorm hp_zero hp_ne_top]
  convert continuous_rpow_const.continuousAt.tendsto.comp this
  simp [zero_rpow_of_pos (_root_.inv_pos.mpr hp)]

Result:
case neg
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ⊤
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s
hi : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
⊢ Tendsto (fun n => ∫⁻ (x : β), ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal ∂μ) atTop (𝓝 0)