Theorem: measurable_limit_of_tendsto_metrizable_ae
File: Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean

tactic: inhabit ι
state_before: α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))

tactic: rcases eq_or_ne L ⊥ with (rfl | hL)
state_before: α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))

tactic: haveI : NeBot L := ⟨hL⟩
state_before: case inr
α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L ≠ ⊥
⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))

tactic: let p : α → (ι → β) → Prop := fun x f' => ∃ l : β, Tendsto (fun n => f' n) L (𝓝 l)
state_before: case inr
α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L ≠ ⊥
this : NeBot L
⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))

tactic: have hp_mem : ∀ x ∈ aeSeqSet hf p, p x fun n => f n x := fun x hx =>
  aeSeq.fun_prop_of_mem_aeSeqSet hf hx
state_before: case inr
α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L ≠ ⊥
this : NeBot L
p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))

tactic: have h_ae_eq : ∀ᵐ x ∂μ, ∀ n, aeSeq hf p n x = f n x := aeSeq.aeSeq_eq_fun_ae hf h_ae_tendsto
state_before: case inr
α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L ≠ ⊥
this : NeBot L
p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x
⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))

tactic: set f_lim : α → β := fun x => dite (x ∈ aeSeqSet hf p) (fun h => (hp_mem x h).choose)
  fun h => (⟨f default x⟩ : Nonempty β).some
state_before: case inr
α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L ≠ ⊥
this : NeBot L
p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x
h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x
⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))

tactic: have h_ae_tendsto_f_lim : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) :=
  h_ae_eq.mono fun x hx => (hf_lim x).congr hx
state_before: case inr
α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L ≠ ⊥
this : NeBot L
p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x
h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x
f_lim : α → β :=
  fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β)
hf_lim : ∀ (x : α), Tendsto (fun n => aeSeq hf p n x) L (𝓝 (f_lim x))
⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))

tactic: have h_f_lim_meas : Measurable f_lim :=
  measurable_of_tendsto_metrizable' L (aeSeq.measurable hf p)
    (tendsto_pi_nhds.mpr fun x => hf_lim x)
state_before: case inr
α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L ≠ ⊥
this : NeBot L
p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x
h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x
f_lim : α → β :=
  fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β)
hf_lim : ∀ (x : α), Tendsto (fun n => aeSeq hf p n x) L (𝓝 (f_lim x))
h_ae_tendsto_f_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))
⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))

tactic: exact ⟨f_lim, h_f_lim_meas, h_ae_tendsto_f_lim⟩
state_before: case inr
α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L ≠ ⊥
this : NeBot L
p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x
h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x
f_lim : α → β :=
  fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β)
hf_lim : ∀ (x : α), Tendsto (fun n => aeSeq hf p n x) L (𝓝 (f_lim x))
h_ae_tendsto_f_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))
h_f_lim_meas : Measurable f_lim
⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))

tactic: exact ⟨(hf default).mk _, (hf default).measurable_mk, eventually_of_forall fun x => tendsto_bot⟩
state_before: case inl
α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
hf : ∀ (n : ι), AEMeasurable (f n)
inhabited_h : Inhabited ι
inst✝ : IsCountablyGenerated ⊥
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) ⊥ (𝓝 l)
⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) ⊥ (𝓝 (f_lim x))

tactic: intro x
state_before: α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L ≠ ⊥
this : NeBot L
p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x
h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x
f_lim : α → β :=
  fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β)
⊢ ∀ (x : α), Tendsto (fun n => aeSeq hf p n x) L (𝓝 (f_lim x))

tactic: simp only [aeSeq]
state_before: α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L ≠ ⊥
this : NeBot L
p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x
h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x
f_lim : α → β :=
  fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β)
x : α
⊢ Tendsto (fun n => aeSeq hf p n x) L (𝓝 (f_lim x))

tactic: split_ifs with h
state_before: α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L ≠ ⊥
this : NeBot L
p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x
h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x
f_lim : α → β :=
  fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β)
x : α
⊢ Tendsto
    (fun n =>
      if x ∈ aeSeqSet hf fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) then
        AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x
      else Nonempty.some (_ : Nonempty β))
    L
    (𝓝
      (if h : x ∈ aeSeqSet hf fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) then
        Exists.choose (_ : p x fun n => f n x)
      else Nonempty.some (_ : Nonempty β)))

tactic: refine' (hp_mem x h).choose_spec.congr fun n => _
state_before: case inl
α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L ≠ ⊥
this : NeBot L
p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x
h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x
f_lim : α → β :=
  fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β)
x : α
h : x ∈ aeSeqSet hf fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
⊢ Tendsto (fun n => AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x) L (𝓝 (Exists.choose (_ : p x fun n => f n x)))

tactic: exact (aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h n).symm
state_before: case inl
α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L ≠ ⊥
this : NeBot L
p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x
h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x
f_lim : α → β :=
  fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β)
x : α
h : x ∈ aeSeqSet hf fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
n : ι
⊢ (fun n => f n x) n = AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x

tactic: exact tendsto_const_nhds
state_before: case inr
α : Type u_2
β : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_1
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : MeasureTheory.Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L ≠ ⊥
this : NeBot L
p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x
h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x
f_lim : α → β :=
  fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β)
x : α
h : ¬x ∈ aeSeqSet hf fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
⊢ Tendsto (fun n => Nonempty.some (_ : Nonempty β)) L (𝓝 (Nonempty.some (_ : Nonempty β)))