Theorem: MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae
File: Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean

Proof steps:
state_before: α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
⊢ ∃ ns, StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
state_after: α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
⊢ ∃ ns, StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
tactic: have h_lt_ε_real : ∀ (ε : ℝ) (_ : 0 < ε), ∃ k : ℕ, 2 * (2 : ℝ)⁻¹ ^ k < ε := by
    intro ε hε
    obtain ⟨k, h_k⟩ : ∃ k : ℕ, (2 : ℝ)⁻¹ ^ k < ε := exists_pow_lt_of_lt_one hε (by norm_num)
    refine ⟨k + 1, (le_of_eq ?_).trans_lt h_k⟩
    rw [pow_add]; ring
state_before: α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
⊢ ∃ ns, StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
state_after: α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
⊢ ∃ ns, StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
tactic: set ns := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
state_before: α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
⊢ ∃ ns, StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
state_after: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
tactic: use ns
state_before: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
state_after: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
tactic: let S := fun k => { x | (2 : ℝ)⁻¹ ^ k ≤ dist (f (ns k) x) (g x) }
state_before: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
state_after: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
tactic: have hμS_le : ∀ k, μ (S k) ≤ (2 : ℝ≥0∞)⁻¹ ^ k :=
    fun k => ExistsSeqTendstoAe.seqTendstoAeSeq_spec hfg k (ns k) le_rfl
state_before: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
state_after: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
tactic: set s := Filter.atTop.limsup S with hs
state_before: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
state_after: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
tactic: have hμs : μ s = 0 := by
    refine measure_limsup_eq_zero (ne_of_lt <| lt_of_le_of_lt (ENNReal.tsum_le_tsum hμS_le) ?_)
    simp only [ENNReal.tsum_geometric, ENNReal.one_sub_inv_two, ENNReal.two_lt_top, inv_inv]
state_before: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
state_after: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
tactic: have h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) := by
    refine fun x hx => Metric.tendsto_atTop.mpr fun ε hε => ?_
    rw [hs, limsup_eq_iInf_iSup_of_nat] at hx
    simp only [S, Set.iSup_eq_iUnion, Set.iInf_eq_iInter, Set.compl_iInter, Set.compl_iUnion,
      Set.mem_iUnion, Set.mem_iInter, Set.mem_compl_iff, Set.mem_setOf_eq, not_le] at hx
    obtain ⟨N, hNx⟩ := hx
    obtain ⟨k, hk_lt_ε⟩ := h_lt_ε_real ε hε
    refine ⟨max N (k - 1), fun n hn_ge => lt_of_le_of_lt ?_ hk_lt_ε⟩
    specialize hNx n ((le_max_left _ _).trans hn_ge)
    have h_inv_n_le_k : (2 : ℝ)⁻¹ ^ n ≤ 2 * (2 : ℝ)⁻¹ ^ k := by
      rw [mul_comm, ← inv_mul_le_iff' (zero_lt_two' ℝ)]
      conv_lhs =>
        congr
        rw [← pow_one (2 : ℝ)⁻¹]
      rw [← pow_add, add_comm]
      exact pow_le_pow_of_le_one (one_div (2 : ℝ) ▸ one_half_pos.le) (inv_le_one one_le_two)
        ((le_tsub_add.trans (add_le_add_right (le_max_right _ _) 1)).trans
          (add_le_add_right hn_ge 1))
    exact le_trans hNx.le h_inv_n_le_k
state_before: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
state_after: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
⊢ StrictMono ns ∧ μ {a | ¬Tendsto (fun i => f (ns i) a) atTop (𝓝 (g a))} = 0
tactic: rw [ae_iff]
state_before: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
⊢ StrictMono ns ∧ μ {a | ¬Tendsto (fun i => f (ns i) a) atTop (𝓝 (g a))} = 0
state_after: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
x : α
⊢ x ∈ {a | ¬Tendsto (fun i => f (ns i) a) atTop (𝓝 (g a))} → x ∈ s
tactic: refine ⟨ExistsSeqTendstoAe.seqTendstoAeSeq_strictMono hfg, measure_mono_null (fun x => ?_) hμs⟩
state_before: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
x : α
⊢ x ∈ {a | ¬Tendsto (fun i => f (ns i) a) atTop (𝓝 (g a))} → x ∈ s
state_after: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
x : α
⊢ x ∉ s → Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
tactic: rw [Set.mem_setOf_eq, ← @Classical.not_not (x ∈ s), not_imp_not]
state_before: case h
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
x : α
⊢ x ∉ s → Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))
state_after: no goals
tactic: exact h_tendsto x