-- Mathlib/MeasureTheory/Function/L1Space/HasFiniteIntegral.lean
theorem all_ae_ofReal_f_le_bound (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
    ∀ᵐ a ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))
⊢ ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)

Tactic:
have F_le_bound := all_ae_ofReal_F_le_bound h_bound

Result:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))
F_le_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a‖ ≤ ENNReal.ofReal (bound a)
⊢ ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))
F_le_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a‖ ≤ ENNReal.ofReal (bound a)
⊢ ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)

Tactic:
rw [← ae_all_iff] at F_le_bound

Result:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))
F_le_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ENNReal.ofReal ‖F i a‖ ≤ ENNReal.ofReal (bound a)
⊢ ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))
F_le_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ENNReal.ofReal ‖F i a‖ ≤ ENNReal.ofReal (bound a)
⊢ ∀ (x : α),
    Tendsto (fun n => ENNReal.ofReal ‖F n x‖) atTop (𝓝 (ENNReal.ofReal ‖f x‖)) →
      (∀ (i : ℕ), ENNReal.ofReal ‖F i x‖ ≤ ENNReal.ofReal (bound x)) → ENNReal.ofReal ‖f x‖ ≤ ENNReal.ofReal (bound x)

Tactic:
intro a tendsto_norm F_le_bound

Result:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))
F_le_bound✝ : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ENNReal.ofReal ‖F i a‖ ≤ ENNReal.ofReal (bound a)
a : α
tendsto_norm : Tendsto (fun n => ENNReal.ofReal ‖F n a‖) atTop (𝓝 (ENNReal.ofReal ‖f a‖))
F_le_bound : ∀ (i : ℕ), ENNReal.ofReal ‖F i a‖ ≤ ENNReal.ofReal (bound a)
⊢ ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))
F_le_bound✝ : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ENNReal.ofReal ‖F i a‖ ≤ ENNReal.ofReal (bound a)
a : α
tendsto_norm : Tendsto (fun n => ENNReal.ofReal ‖F n a‖) atTop (𝓝 (ENNReal.ofReal ‖f a‖))
F_le_bound : ∀ (i : ℕ), ENNReal.ofReal ‖F i a‖ ≤ ENNReal.ofReal (bound a)
⊢ ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)

Tactic:
exact le_of_tendsto' tendsto_norm F_le_bound

Result:
no goals

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))
F_le_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ENNReal.ofReal ‖F i a‖ ≤ ENNReal.ofReal (bound a)
⊢ ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)

Tactic:
apply F_le_bound.mp ((all_ae_tendsto_ofReal_norm h_lim).mono _)

Result:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))
F_le_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ENNReal.ofReal ‖F i a‖ ≤ ENNReal.ofReal (bound a)
⊢ ∀ (x : α),
    Tendsto (fun n => ENNReal.ofReal ‖F n x‖) atTop (𝓝 (ENNReal.ofReal ‖f x‖)) →
      (∀ (i : ℕ), ENNReal.ofReal ‖F i x‖ ≤ ENNReal.ofReal (bound x)) → ENNReal.ofReal ‖f x‖ ≤ ENNReal.ofReal (bound x)