Theorem: FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto
File: Mathlib/Topology/Sequences.lean

tactic: refine ⟨fun s x hcx => ?_⟩
state_before: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
⊢ FrechetUrysohnSpace X
state_after: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure s
⊢ x ∈ seqClosure s

tactic: by_cases hx : x ∈ s
state_before: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure s
⊢ x ∈ seqClosure s
state_after: case pos
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure s
hx : x ∈ s
⊢ x ∈ seqClosure s

case neg
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure s
hx : x ∉ s
⊢ x ∈ seqClosure s

tactic: exact subset_seqClosure hx
state_before: case pos
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure s
hx : x ∈ s
⊢ x ∈ seqClosure s
state_after: no goals

tactic: rcases extraction_of_frequently_atTop hus with ⟨φ, φ_mono, hφ⟩
state_before: case neg.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure s
hx : x ∉ s
u : ℕ → X
hux : Tendsto u atTop (𝓝 x)
hus : ∃ᶠ (x : ℕ) in atTop, u x ∈ s
⊢ x ∈ seqClosure s
state_after: case neg.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure s
hx : x ∉ s
u : ℕ → X
hux : Tendsto u atTop (𝓝 x)
hus : ∃ᶠ (x : ℕ) in atTop, u x ∈ s
φ : ℕ → ℕ
φ_mono : StrictMono φ
hφ : ∀ (n : ℕ), u (φ n) ∈ s
⊢ x ∈ seqClosure s

tactic: exact ⟨u ∘ φ, hφ, hux.comp φ_mono.tendsto_atTop⟩
state_before: case neg.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure s
hx : x ∉ s
u : ℕ → X
hux : Tendsto u atTop (𝓝 x)
hus : ∃ᶠ (x : ℕ) in atTop, u x ∈ s
φ : ℕ → ℕ
φ_mono : StrictMono φ
hφ : ∀ (n : ℕ), u (φ n) ∈ s
⊢ x ∈ seqClosure s
state_after: no goals