-- Mathlib/MeasureTheory/Function/Egorov.lean
theorem measure_iUnionNotConvergentSeq (hε : 0 < ε) (hf : ∀ n, StronglyMeasurable (f n))
    (hg : StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ∞)
    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
    μ (iUnionNotConvergentSeq hε hf hg hsm hs hfg) ≤ ENNReal.ofReal ε

Proof state:
case hf_nonneg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : MetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
n : ℕ
⊢ 0 ≤ 2⁻¹ ^ n

Tactic:
positivity

Result:
no goals

Proof state:
case hf
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : MetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ Summable (HPow.hPow (1 / 2))

Tactic:
exact summable_geometric_two

Result:
no goals

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : MetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ μ (iUnionNotConvergentSeq hε hf hg hsm hs hfg) ≤ ENNReal.ofReal ε

Tactic:
refine le_trans (measure_iUnion_le _) (le_trans
  (ENNReal.tsum_le_tsum <| notConvergentSeqLTIndex_spec (half_pos hε) hf hg hsm hs hfg) ?_)

Result:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : MetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ ∑' (a : ℕ), ENNReal.ofReal (ε / 2 * 2⁻¹ ^ a) ≤ ENNReal.ofReal ε

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : MetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ ∑' (a : ℕ), ENNReal.ofReal (ε / 2 * 2⁻¹ ^ a) ≤ ENNReal.ofReal ε

Tactic:
simp_rw [ENNReal.ofReal_mul (half_pos hε).le]

Result:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : MetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ ∑' (a : ℕ), ENNReal.ofReal (ε / 2) * ENNReal.ofReal (2⁻¹ ^ a) ≤ ENNReal.ofReal ε

Proof state:
case hf_nonneg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : MetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ ∀ (n : ℕ), 0 ≤ 2⁻¹ ^ n

Tactic:
intro n

Result:
case hf_nonneg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : MetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
n : ℕ
⊢ 0 ≤ 2⁻¹ ^ n

Proof state:
case hf
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : MetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ Summable (HPow.hPow 2⁻¹)

Tactic:
rw [inv_eq_one_div]

Result:
case hf
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : MetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ Summable (HPow.hPow (1 / 2))

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : MetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ ∑' (a : ℕ), ENNReal.ofReal (ε / 2) * ENNReal.ofReal (2⁻¹ ^ a) ≤ ENNReal.ofReal ε

Tactic:
rw [ENNReal.tsum_mul_left, ← ENNReal.ofReal_tsum_of_nonneg, inv_eq_one_div, tsum_geometric_two,
  ← ENNReal.ofReal_mul (half_pos hε).le, div_mul_cancel₀ ε two_ne_zero]

Result:
case hf_nonneg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : MetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ ∀ (n : ℕ), 0 ≤ 2⁻¹ ^ n

case hf
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : MetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ Summable (HPow.hPow 2⁻¹)