-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
theorem exists_null_frontiers_thickening (μ : Measure Ω) [SFinite μ] (s : Set Ω) :
    ∃ rs : ℕ → ℝ,
      Tendsto rs atTop (𝓝 0) ∧ ∀ n, 0 < rs n ∧ μ (frontier (Metric.thickening (rs n) s)) = 0

Proof state:
case intro.intro.intro
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
Rs : ℕ → ℝ
left✝ : StrictAnti Rs
Rs_pos : ∀ (n : ℕ), 0 < Rs n
Rs_lim : Tendsto Rs atTop (𝓝 0)
obs : ∀ (n : ℕ), ∃ r ∈ Ioo 0 (Rs n), μ (frontier (Metric.thickening r s)) = 0
⊢ ∃ rs, Tendsto rs atTop (𝓝 0) ∧ ∀ (n : ℕ), 0 < rs n ∧ μ (frontier (Metric.thickening (rs n) s)) = 0

Tactic:
refine ⟨fun n : ℕ => (obs n).choose, ⟨?_, ?_⟩⟩

Result:
case intro.intro.intro.refine_1
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
Rs : ℕ → ℝ
left✝ : StrictAnti Rs
Rs_pos : ∀ (n : ℕ), 0 < Rs n
Rs_lim : Tendsto Rs atTop (𝓝 0)
obs : ∀ (n : ℕ), ∃ r ∈ Ioo 0 (Rs n), μ (frontier (Metric.thickening r s)) = 0
⊢ Tendsto (fun n => ⋯.choose) atTop (𝓝 0)

case intro.intro.intro.refine_2
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
Rs : ℕ → ℝ
left✝ : StrictAnti Rs
Rs_pos : ∀ (n : ℕ), 0 < Rs n
Rs_lim : Tendsto Rs atTop (𝓝 0)
obs : ∀ (n : ℕ), ∃ r ∈ Ioo 0 (Rs n), μ (frontier (Metric.thickening r s)) = 0
⊢ ∀ (n : ℕ), 0 < (fun n => ⋯.choose) n ∧ μ (frontier (Metric.thickening ((fun n => ⋯.choose) n) s)) = 0

Proof state:
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
⊢ ∃ rs, Tendsto rs atTop (𝓝 0) ∧ ∀ (n : ℕ), 0 < rs n ∧ μ (frontier (Metric.thickening (rs n) s)) = 0

Tactic:
rcases exists_seq_strictAnti_tendsto (0 : ℝ) with ⟨Rs, ⟨_, ⟨Rs_pos, Rs_lim⟩⟩⟩

Result:
case intro.intro.intro
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
Rs : ℕ → ℝ
left✝ : StrictAnti Rs
Rs_pos : ∀ (n : ℕ), 0 < Rs n
Rs_lim : Tendsto Rs atTop (𝓝 0)
⊢ ∃ rs, Tendsto rs atTop (𝓝 0) ∧ ∀ (n : ℕ), 0 < rs n ∧ μ (frontier (Metric.thickening (rs n) s)) = 0

Proof state:
case intro.intro.intro.refine_1
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
Rs : ℕ → ℝ
left✝ : StrictAnti Rs
Rs_pos : ∀ (n : ℕ), 0 < Rs n
Rs_lim : Tendsto Rs atTop (𝓝 0)
obs : ∀ (n : ℕ), ∃ r ∈ Ioo 0 (Rs n), μ (frontier (Metric.thickening r s)) = 0
⊢ Tendsto (fun n => ⋯.choose) atTop (𝓝 0)

Tactic:
exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds Rs_lim
  (fun n ↦ (obs n).choose_spec.1.1.le) fun n ↦ (obs n).choose_spec.1.2.le

Result:
no goals

Proof state:
case intro.intro.intro.refine_2
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
Rs : ℕ → ℝ
left✝ : StrictAnti Rs
Rs_pos : ∀ (n : ℕ), 0 < Rs n
Rs_lim : Tendsto Rs atTop (𝓝 0)
obs : ∀ (n : ℕ), ∃ r ∈ Ioo 0 (Rs n), μ (frontier (Metric.thickening r s)) = 0
⊢ ∀ (n : ℕ), 0 < (fun n => ⋯.choose) n ∧ μ (frontier (Metric.thickening ((fun n => ⋯.choose) n) s)) = 0

Tactic:
exact fun n ↦ ⟨(obs n).choose_spec.1.1, (obs n).choose_spec.2⟩

Result:
no goals

Proof state:
case intro.intro.intro
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
Rs : ℕ → ℝ
left✝ : StrictAnti Rs
Rs_pos : ∀ (n : ℕ), 0 < Rs n
Rs_lim : Tendsto Rs atTop (𝓝 0)
⊢ ∃ rs, Tendsto rs atTop (𝓝 0) ∧ ∀ (n : ℕ), 0 < rs n ∧ μ (frontier (Metric.thickening (rs n) s)) = 0

Tactic:
have obs := fun n : ℕ => exists_null_frontier_thickening μ s (Rs_pos n)

Result:
case intro.intro.intro
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
Rs : ℕ → ℝ
left✝ : StrictAnti Rs
Rs_pos : ∀ (n : ℕ), 0 < Rs n
Rs_lim : Tendsto Rs atTop (𝓝 0)
obs : ∀ (n : ℕ), ∃ r ∈ Ioo 0 (Rs n), μ (frontier (Metric.thickening r s)) = 0
⊢ ∃ rs, Tendsto rs atTop (𝓝 0) ∧ ∀ (n : ℕ), 0 < rs n ∧ μ (frontier (Metric.thickening (rs n) s)) = 0