l3lab/ntp-mathlib huggingface dataset:
/-- Given an antitone basis `s : ℕ → Set α` of a filter, extract an antitone subbasis `s ∘ φ`,
`φ : ℕ → ℕ`, such that `m < n` implies `r (φ m) (φ n)`. This lemma can be used to extract an
antitone basis with basis sets decreasing "sufficiently fast". -/
theorem HasAntitoneBasis.subbasis_with_rel {f : Filter α} {s : ℕ → Set α}
    (hs : f.HasAntitoneBasis s) {r : ℕ → ℕ → Prop} (hr : ∀ m, ∀ᶠ n in atTop, r m n) :
    ∃ φ : ℕ → ℕ, StrictMono φ ∧ (∀ ⦃m n⦄, m < n → r (φ m) (φ n)) ∧ f.HasAntitoneBasis (s ∘ φ) := by
  rsuffices ⟨φ, hφ, hrφ⟩ : ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ m n, m < n → r (φ m) (φ n)
  · exact ⟨φ, hφ, hrφ, hs.comp_strictMono hφ⟩
  have : ∀ t : Set ℕ, t.Finite → ∀ᶠ n in atTop, ∀ m ∈ t, m < n ∧ r m n := fun t ht =>
    (eventually_all_finite ht).2 fun m _ => (eventually_gt_atTop m).and (hr _)
  rcases seq_of_forall_finite_exists fun t ht => (this t ht).exists with ⟨φ, hφ⟩
  simp only [ball_image_iff, forall_and, mem_Iio] at hφ
  

Proof state:
case intro
ι : Type u_1
ι' : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
f : Filter α
s : ℕ → Set α
hs : HasAntitoneBasis f s
r : ℕ → ℕ → Prop
hr : ∀ (m : ℕ), ∀ᶠ (n : ℕ) in atTop, r m n
this : ∀ (t : Set ℕ), Set.Finite t → ∀ᶠ (n : ℕ) in atTop, ∀ m ∈ t, m < n ∧ r m n
φ : ℕ → ℕ
hφ : (∀ (x x_1 : ℕ), x_1 < x → φ x_1 < φ x) ∧ ∀ (x x_1 : ℕ), x_1 < x → r (φ x_1) (φ x)
⊢ ∃ φ, StrictMono φ ∧ ∀ (m n : ℕ), m < n → r (φ m) (φ n)

Next tactic:
exact ⟨φ, forall_swap.2 hφ.1, forall_swap.2 hφ.2⟩
/-- If `f` is a nontrivial countably generated filter, then there exists a sequence that converges
to `f`. -/
theorem exists_seq_tendsto (f : Filter α) [IsCountablyGenerated f] [NeBot f] :
    ∃ x : ℕ → α, Tendsto x atTop f := by
  

Proof state:
ι : Type u_1
ι' : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
f : Filter α
inst✝¹ : IsCountablyGenerated f
inst✝ : NeBot f
⊢ ∃ x, Tendsto x atTop f

Next tactic:
obtain ⟨B, h⟩ := f.exists_antitone_basis
/-- If `f` is a nontrivial countably generated filter, then there exists a sequence that converges
to `f`. -/
theorem exists_seq_tendsto (f : Filter α) [IsCountablyGenerated f] [NeBot f] :
    ∃ x : ℕ → α, Tendsto x atTop f := by
  obtain ⟨B, h⟩ := f.exists_antitone_basis
  

Proof state:
case intro
ι : Type u_1
ι' : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
f : Filter α
inst✝¹ : IsCountablyGenerated f
inst✝ : NeBot f
B : ℕ → Set α
h : HasAntitoneBasis f B
⊢ ∃ x, Tendsto x atTop f

Next tactic:
choose x hx using fun n => Filter.nonempty_of_mem (h.mem n)
/-- If `f` is a nontrivial countably generated filter, then there exists a sequence that converges
to `f`. -/
theorem exists_seq_tendsto (f : Filter α) [IsCountablyGenerated f] [NeBot f] :
    ∃ x : ℕ → α, Tendsto x atTop f := by
  obtain ⟨B, h⟩ := f.exists_antitone_basis
  choose x hx using fun n => Filter.nonempty_of_mem (h.mem n)
  

Proof state:
case intro
ι : Type u_1
ι' : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
f : Filter α
inst✝¹ : IsCountablyGenerated f
inst✝ : NeBot f
B : ℕ → Set α
h : HasAntitoneBasis f B
x : ℕ → α
hx : ∀ (n : ℕ), x n ∈ B n
⊢ ∃ x, Tendsto x atTop f

Next tactic:
exact ⟨x, h.tendsto hx⟩
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  

Proof state:
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
⊢ ∃ xs, Monotone xs ∧ Tendsto xs atTop atTop

Next tactic:
haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  

Proof state:
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
⊢ ∃ xs, Monotone xs ∧ Tendsto xs atTop atTop

Next tactic:
obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  

Proof state:
case intro
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : Tendsto ys atTop atTop
⊢ ∃ xs, Monotone xs ∧ Tendsto xs atTop atTop

Next tactic:
let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  

Proof state:
case intro
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : Tendsto ys atTop atTop
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
⊢ ∃ xs, Monotone xs ∧ Tendsto xs atTop atTop

Next tactic:
have h_mono : Monotone xs := by
    intro i j hij
    rw [Finset.sup'_le_iff]
    intro k hk
    refine' Finset.le_sup'_of_le _ _ le_rfl
    rw [Finset.mem_range] at hk ⊢
    exact hk.trans_le (add_le_add_right hij _)
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    

Proof state:
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : Tendsto ys atTop atTop
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
⊢ Monotone xs

Next tactic:
intro i j hij
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    intro i j hij
    

Proof state:
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : Tendsto ys atTop atTop
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
i j : ℕ
hij : i ≤ j
⊢ xs i ≤ xs j

Next tactic:
rw [Finset.sup'_le_iff]
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    intro i j hij
    rw [Finset.sup'_le_iff]
    

Proof state:
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : Tendsto ys atTop atTop
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
i j : ℕ
hij : i ≤ j
⊢ ∀ b ∈ Finset.range (i + 1), ys b ≤ xs j

Next tactic:
intro k hk
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    intro i j hij
    rw [Finset.sup'_le_iff]
    intro k hk
    

Proof state:
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : Tendsto ys atTop atTop
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
i j : ℕ
hij : i ≤ j
k : ℕ
hk : k ∈ Finset.range (i + 1)
⊢ ys k ≤ xs j

Next tactic:
refine' Finset.le_sup'_of_le _ _ le_rfl
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    intro i j hij
    rw [Finset.sup'_le_iff]
    intro k hk
    refine' Finset.le_sup'_of_le _ _ le_rfl
    

Proof state:
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : Tendsto ys atTop atTop
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
i j : ℕ
hij : i ≤ j
k : ℕ
hk : k ∈ Finset.range (i + 1)
⊢ k ∈ Finset.range (j + 1)

Next tactic:
rw [Finset.mem_range] at hk ⊢
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    intro i j hij
    rw [Finset.sup'_le_iff]
    intro k hk
    refine' Finset.le_sup'_of_le _ _ le_rfl
    rw [Finset.mem_range] at hk ⊢
    

Proof state:
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : Tendsto ys atTop atTop
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
i j : ℕ
hij : i ≤ j
k : ℕ
hk : k < i + 1
⊢ k < j + 1

Next tactic:
exact hk.trans_le (add_le_add_right hij _)
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    intro i j hij
    rw [Finset.sup'_le_iff]
    intro k hk
    refine' Finset.le_sup'_of_le _ _ le_rfl
    rw [Finset.mem_range] at hk ⊢
    exact hk.trans_le (add_le_add_right hij _)
  

Proof state:
case intro
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : Tendsto ys atTop atTop
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
h_mono : Monotone xs
⊢ ∃ xs, Monotone xs ∧ Tendsto xs atTop atTop

Next tactic:
refine' ⟨xs, h_mono, _⟩
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    intro i j hij
    rw [Finset.sup'_le_iff]
    intro k hk
    refine' Finset.le_sup'_of_le _ _ le_rfl
    rw [Finset.mem_range] at hk ⊢
    exact hk.trans_le (add_le_add_right hij _)
  refine' ⟨xs, h_mono, _⟩
  · 

Proof state:
case intro
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : Tendsto ys atTop atTop
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
h_mono : Monotone xs
⊢ Tendsto xs atTop atTop

Next tactic:
refine' tendsto_atTop_atTop_of_monotone h_mono _
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    intro i j hij
    rw [Finset.sup'_le_iff]
    intro k hk
    refine' Finset.le_sup'_of_le _ _ le_rfl
    rw [Finset.mem_range] at hk ⊢
    exact hk.trans_le (add_le_add_right hij _)
  refine' ⟨xs, h_mono, _⟩
  · refine' tendsto_atTop_atTop_of_monotone h_mono _
    

Proof state:
case intro
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : Tendsto ys atTop atTop
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
h_mono : Monotone xs
⊢ ∀ (b : α), ∃ a, b ≤ xs a

Next tactic:
have : ∀ a : α, ∃ n : ℕ, a ≤ ys n := by
      rw [tendsto_atTop_atTop] at h
      intro a
      obtain ⟨i, hi⟩ := h a
      exact ⟨i, hi i le_rfl⟩
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    intro i j hij
    rw [Finset.sup'_le_iff]
    intro k hk
    refine' Finset.le_sup'_of_le _ _ le_rfl
    rw [Finset.mem_range] at hk ⊢
    exact hk.trans_le (add_le_add_right hij _)
  refine' ⟨xs, h_mono, _⟩
  · refine' tendsto_atTop_atTop_of_monotone h_mono _
    have : ∀ a : α, ∃ n : ℕ, a ≤ ys n := by
      

Proof state:
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : Tendsto ys atTop atTop
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
h_mono : Monotone xs
⊢ ∀ (a : α), ∃ n, a ≤ ys n

Next tactic:
rw [tendsto_atTop_atTop] at h
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    intro i j hij
    rw [Finset.sup'_le_iff]
    intro k hk
    refine' Finset.le_sup'_of_le _ _ le_rfl
    rw [Finset.mem_range] at hk ⊢
    exact hk.trans_le (add_le_add_right hij _)
  refine' ⟨xs, h_mono, _⟩
  · refine' tendsto_atTop_atTop_of_monotone h_mono _
    have : ∀ a : α, ∃ n : ℕ, a ≤ ys n := by
      rw [tendsto_atTop_atTop] at h
      

Proof state:
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : ∀ (b : α), ∃ i, ∀ (a : ℕ), i ≤ a → b ≤ ys a
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
h_mono : Monotone xs
⊢ ∀ (a : α), ∃ n, a ≤ ys n

Next tactic:
intro a
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    intro i j hij
    rw [Finset.sup'_le_iff]
    intro k hk
    refine' Finset.le_sup'_of_le _ _ le_rfl
    rw [Finset.mem_range] at hk ⊢
    exact hk.trans_le (add_le_add_right hij _)
  refine' ⟨xs, h_mono, _⟩
  · refine' tendsto_atTop_atTop_of_monotone h_mono _
    have : ∀ a : α, ∃ n : ℕ, a ≤ ys n := by
      rw [tendsto_atTop_atTop] at h
      intro a
      

Proof state:
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : ∀ (b : α), ∃ i, ∀ (a : ℕ), i ≤ a → b ≤ ys a
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
h_mono : Monotone xs
a : α
⊢ ∃ n, a ≤ ys n

Next tactic:
obtain ⟨i, hi⟩ := h a
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    intro i j hij
    rw [Finset.sup'_le_iff]
    intro k hk
    refine' Finset.le_sup'_of_le _ _ le_rfl
    rw [Finset.mem_range] at hk ⊢
    exact hk.trans_le (add_le_add_right hij _)
  refine' ⟨xs, h_mono, _⟩
  · refine' tendsto_atTop_atTop_of_monotone h_mono _
    have : ∀ a : α, ∃ n : ℕ, a ≤ ys n := by
      rw [tendsto_atTop_atTop] at h
      intro a
      obtain ⟨i, hi⟩ := h a
      

Proof state:
case intro
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : ∀ (b : α), ∃ i, ∀ (a : ℕ), i ≤ a → b ≤ ys a
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
h_mono : Monotone xs
a : α
i : ℕ
hi : ∀ (a_1 : ℕ), i ≤ a_1 → a ≤ ys a_1
⊢ ∃ n, a ≤ ys n

Next tactic:
exact ⟨i, hi i le_rfl⟩
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    intro i j hij
    rw [Finset.sup'_le_iff]
    intro k hk
    refine' Finset.le_sup'_of_le _ _ le_rfl
    rw [Finset.mem_range] at hk ⊢
    exact hk.trans_le (add_le_add_right hij _)
  refine' ⟨xs, h_mono, _⟩
  · refine' tendsto_atTop_atTop_of_monotone h_mono _
    have : ∀ a : α, ∃ n : ℕ, a ≤ ys n := by
      rw [tendsto_atTop_atTop] at h
      intro a
      obtain ⟨i, hi⟩ := h a
      exact ⟨i, hi i le_rfl⟩
    

Proof state:
case intro
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : Tendsto ys atTop atTop
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
h_mono : Monotone xs
this : ∀ (a : α), ∃ n, a ≤ ys n
⊢ ∀ (b : α), ∃ a, b ≤ xs a

Next tactic:
intro a
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    intro i j hij
    rw [Finset.sup'_le_iff]
    intro k hk
    refine' Finset.le_sup'_of_le _ _ le_rfl
    rw [Finset.mem_range] at hk ⊢
    exact hk.trans_le (add_le_add_right hij _)
  refine' ⟨xs, h_mono, _⟩
  · refine' tendsto_atTop_atTop_of_monotone h_mono _
    have : ∀ a : α, ∃ n : ℕ, a ≤ ys n := by
      rw [tendsto_atTop_atTop] at h
      intro a
      obtain ⟨i, hi⟩ := h a
      exact ⟨i, hi i le_rfl⟩
    intro a
    

Proof state:
case intro
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : Tendsto ys atTop atTop
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
h_mono : Monotone xs
this : ∀ (a : α), ∃ n, a ≤ ys n
a : α
⊢ ∃ a_1, a ≤ xs a_1

Next tactic:
obtain ⟨i, hi⟩ := this a
theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
    [(atTop : Filter α).IsCountablyGenerated] :
    ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
  haveI h_ne_bot : (atTop : Filter α).NeBot := atTop_neBot
  obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
  have h_mono : Monotone xs := by
    intro i j hij
    rw [Finset.sup'_le_iff]
    intro k hk
    refine' Finset.le_sup'_of_le _ _ le_rfl
    rw [Finset.mem_range] at hk ⊢
    exact hk.trans_le (add_le_add_right hij _)
  refine' ⟨xs, h_mono, _⟩
  · refine' tendsto_atTop_atTop_of_monotone h_mono _
    have : ∀ a : α, ∃ n : ℕ, a ≤ ys n := by
      rw [tendsto_atTop_atTop] at h
      intro a
      obtain ⟨i, hi⟩ := h a
      exact ⟨i, hi i le_rfl⟩
    intro a
    obtain ⟨i, hi⟩ := this a
    

Proof state:
case intro.intro
ι : Type u_1
ι' : Type u_2
α✝ : Type u_3
β : Type u_4
γ : Type u_5
α : Type u_6
inst✝² : SemilatticeSup α
inst✝¹ : Nonempty α
inst✝ : IsCountablyGenerated atTop
h_ne_bot : NeBot atTop
ys : ℕ → α
h : Tendsto ys atTop atTop
xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys
h_mono : Monotone xs
this : ∀ (a : α), ∃ n, a ≤ ys n
a : α
i : ℕ
hi : a ≤ ys i
⊢ ∃ a_1, a ≤ xs a_1

Next tactic:
refine' ⟨i, hi.trans _⟩