Theorem: ProbabilityTheory.IsMeasurableRatCDF.tendsto_stieltjesFunction_atBot
File: Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
⊢ Tendsto (↑(hf.stieltjesFunction a)) atBot (𝓝 0)
state_after: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
h_exists : ∀ (x : ℝ), ∃ q, x < ↑q ∧ ↑q < x + 1
⊢ Tendsto (↑(hf.stieltjesFunction a)) atBot (𝓝 0)
tactic: have h_exists : ∀ x : ℝ, ∃ q : ℚ, x < q ∧ ↑q < x + 1 := fun x ↦ exists_rat_btwn (lt_add_one x)
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
h_exists : ∀ (x : ℝ), ∃ q, x < ↑q ∧ ↑q < x + 1
⊢ Tendsto (↑(hf.stieltjesFunction a)) atBot (𝓝 0)
state_after: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
h_exists : ∀ (x : ℝ), ∃ q, x < ↑q ∧ ↑q < x + 1
qs : ℝ → ℚ := fun x => ⋯.choose
⊢ Tendsto (↑(hf.stieltjesFunction a)) atBot (𝓝 0)
tactic: let qs : ℝ → ℚ := fun x ↦ (h_exists x).choose
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
h_exists : ∀ (x : ℝ), ∃ q, x < ↑q ∧ ↑q < x + 1
qs : ℝ → ℚ := fun x => ⋯.choose
⊢ Tendsto (↑(hf.stieltjesFunction a)) atBot (𝓝 0)
state_after: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
h_exists : ∀ (x : ℝ), ∃ q, x < ↑q ∧ ↑q < x + 1
qs : ℝ → ℚ := fun x => ⋯.choose
hqs_tendsto : Tendsto qs atBot atBot
⊢ Tendsto (↑(hf.stieltjesFunction a)) atBot (𝓝 0)
tactic: have hqs_tendsto : Tendsto qs atBot atBot := by
    rw [tendsto_atBot_atBot]
    refine fun q ↦ ⟨q - 1, fun y hy ↦ ?_⟩
    have h_le : ↑(qs y) ≤ (q : ℝ) - 1 + 1 :=
      (h_exists y).choose_spec.2.le.trans (add_le_add hy le_rfl)
    rw [sub_add_cancel] at h_le
    exact mod_cast h_le
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
h_exists : ∀ (x : ℝ), ∃ q, x < ↑q ∧ ↑q < x + 1
qs : ℝ → ℚ := fun x => ⋯.choose
hqs_tendsto : Tendsto qs atBot atBot
⊢ Tendsto (↑(hf.stieltjesFunction a)) atBot (𝓝 0)
state_after: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
h_exists : ∀ (x : ℝ), ∃ q, x < ↑q ∧ ↑q < x + 1
qs : ℝ → ℚ := fun x => ⋯.choose
hqs_tendsto : Tendsto qs atBot atBot
x : ℝ
⊢ ↑(hf.stieltjesFunction a) x ≤ (f a ∘ qs) x
tactic: refine tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
    ((hf.tendsto_atBot_zero a).comp hqs_tendsto) (stieltjesFunction_nonneg hf a) fun x ↦ ?_
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
h_exists : ∀ (x : ℝ), ∃ q, x < ↑q ∧ ↑q < x + 1
qs : ℝ → ℚ := fun x => ⋯.choose
hqs_tendsto : Tendsto qs atBot atBot
x : ℝ
⊢ ↑(hf.stieltjesFunction a) x ≤ (f a ∘ qs) x
state_after: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
h_exists : ∀ (x : ℝ), ∃ q, x < ↑q ∧ ↑q < x + 1
qs : ℝ → ℚ := fun x => ⋯.choose
hqs_tendsto : Tendsto qs atBot atBot
x : ℝ
⊢ ↑(hf.stieltjesFunction a) x ≤ ↑(hf.stieltjesFunction a) ↑(qs x)
tactic: rw [Function.comp_apply, ← stieltjesFunction_eq hf]
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
h_exists : ∀ (x : ℝ), ∃ q, x < ↑q ∧ ↑q < x + 1
qs : ℝ → ℚ := fun x => ⋯.choose
hqs_tendsto : Tendsto qs atBot atBot
x : ℝ
⊢ ↑(hf.stieltjesFunction a) x ≤ ↑(hf.stieltjesFunction a) ↑(qs x)
state_after: no goals
tactic: exact (hf.stieltjesFunction a).mono (h_exists x).choose_spec.1.le