case pos
α : Type u
β : Type v
γ : Type w
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : LinearOrder α
inst✝³ : LinearOrder β
inst✝² : OrderTopology α
inst✝¹ : OrderTopology β
t : Set α
x : α
inst✝ : IsCountablyGenerated (𝓝 x)
htx : IsLUB t x
ht : Set.Nonempty t
h : x ∈ t
⊢ ∃ u, Monotone u ∧ (∀ (n : ℕ), u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t

case neg
α : Type u
β : Type v
γ : Type w
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : LinearOrder α
inst✝³ : LinearOrder β
inst✝² : OrderTopology α
inst✝¹ : OrderTopology β
t : Set α
x : α
inst✝ : IsCountablyGenerated (𝓝 x)
htx : IsLUB t x
ht : Set.Nonempty t
h : ¬x ∈ t
⊢ ∃ u, Monotone u ∧ (∀ (n : ℕ), u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t
α : Type u
β : Type v
γ : Type w
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : LinearOrder α
inst✝³ : LinearOrder β
inst✝² : OrderTopology α
inst✝¹ : OrderTopology β
t : Set α
x : α
inst✝ : IsCountablyGenerated (𝓝 x)
htx : IsLUB t x
ht : Set.Nonempty t
⊢ ∃ u, Monotone u ∧ (∀ (n : ℕ), u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t
by_cases h : x ∈ t
no goals
case pos
α : Type u
β : Type v
γ : Type w
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : LinearOrder α
inst✝³ : LinearOrder β
inst✝² : OrderTopology α
inst✝¹ : OrderTopology β
t : Set α
x : α
inst✝ : IsCountablyGenerated (𝓝 x)
htx : IsLUB t x
ht : Set.Nonempty t
h : x ∈ t
⊢ ∃ u, Monotone u ∧ (∀ (n : ℕ), u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t
exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩
case neg.intro
α : Type u
β : Type v
γ : Type w
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : LinearOrder α
inst✝³ : LinearOrder β
inst✝² : OrderTopology α
inst✝¹ : OrderTopology β
t : Set α
x : α
inst✝ : IsCountablyGenerated (𝓝 x)
htx : IsLUB t x
ht : Set.Nonempty t
h : ¬x ∈ t
u : ℕ → α
hu : StrictMono u ∧ (∀ (n : ℕ), u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t
⊢ ∃ u, Monotone u ∧ (∀ (n : ℕ), u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t
case neg
α : Type u
β : Type v
γ : Type w
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : LinearOrder α
inst✝³ : LinearOrder β
inst✝² : OrderTopology α
inst✝¹ : OrderTopology β
t : Set α
x : α
inst✝ : IsCountablyGenerated (𝓝 x)
htx : IsLUB t x
ht : Set.Nonempty t
h : ¬x ∈ t
⊢ ∃ u, Monotone u ∧ (∀ (n : ℕ), u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t
rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩
no goals
case neg.intro
α : Type u
β : Type v
γ : Type w
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : LinearOrder α
inst✝³ : LinearOrder β
inst✝² : OrderTopology α
inst✝¹ : OrderTopology β
t : Set α
x : α
inst✝ : IsCountablyGenerated (𝓝 x)
htx : IsLUB t x
ht : Set.Nonempty t
h : ¬x ∈ t
u : ℕ → α
hu : StrictMono u ∧ (∀ (n : ℕ), u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t
⊢ ∃ u, Monotone u ∧ (∀ (n : ℕ), u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t
exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩