Theorem: iUnion_Ici_eq_Ioi_of_lt_of_tendsto
File: Mathlib/Topology/Algebra/Order/LiminfLimsup.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
S : Type u_5
π : ι → Type u_6
inst✝³ : CompleteLinearOrder R
inst✝² : TopologicalSpace R
inst✝¹ : OrderTopology R
x : R
as : ι → R
x_lt : ∀ (i : ι), x < as i
F : Filter ι
inst✝ : F.NeBot
as_lim : Tendsto as F (𝓝 x)
⊢ ⋃ i, Ici (as i) = Ioi x
state_after: ι : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
S : Type u_5
π : ι → Type u_6
inst✝³ : CompleteLinearOrder R
inst✝² : TopologicalSpace R
inst✝¹ : OrderTopology R
x : R
as : ι → R
x_lt : ∀ (i : ι), x < as i
F : Filter ι
inst✝ : F.NeBot
as_lim : Tendsto as F (𝓝 x)
obs : x ∉ range as
⊢ ⋃ i, Ici (as i) = Ioi x
tactic: have obs : x ∉ range as := by
    intro maybe_x_is
    rcases mem_range.mp maybe_x_is with ⟨i, hi⟩
    simpa only [hi, lt_self_iff_false] using x_lt i
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
S : Type u_5
π : ι → Type u_6
inst✝³ : CompleteLinearOrder R
inst✝² : TopologicalSpace R
inst✝¹ : OrderTopology R
x : R
as : ι → R
x_lt : ∀ (i : ι), x < as i
F : Filter ι
inst✝ : F.NeBot
as_lim : Tendsto as F (𝓝 x)
obs : x ∉ range as
⊢ ⋃ i, Ici (as i) = Ioi x
state_after: ι : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
S : Type u_5
π : ι → Type u_6
inst✝³ : CompleteLinearOrder R
inst✝² : TopologicalSpace R
inst✝¹ : OrderTopology R
x : R
as : ι → R
x_lt : ∀ (i : ι), x < as i
F : Filter ι
inst✝ : F.NeBot
as_lim : Tendsto as F (𝓝 x)
obs : x ∉ range as
this : ⨅ i, as i = x
⊢ ⋃ i, Ici (as i) = Ioi x
tactic: have := iInf_eq_of_forall_le_of_tendsto (fun i ↦ (x_lt i).le) as_lim
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
S : Type u_5
π : ι → Type u_6
inst✝³ : CompleteLinearOrder R
inst✝² : TopologicalSpace R
inst✝¹ : OrderTopology R
x : R
as : ι → R
x_lt : ∀ (i : ι), x < as i
F : Filter ι
inst✝ : F.NeBot
as_lim : Tendsto as F (𝓝 x)
obs : x ∉ range as
this : ⨅ i, as i = x
⊢ ⋃ i, Ici (as i) = Ioi x
state_after: ι : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
S : Type u_5
π : ι → Type u_6
inst✝³ : CompleteLinearOrder R
inst✝² : TopologicalSpace R
inst✝¹ : OrderTopology R
x : R
as : ι → R
x_lt : ∀ (i : ι), x < as i
F : Filter ι
inst✝ : F.NeBot
as_lim : Tendsto as F (𝓝 x)
obs : ⨅ i, as i ∉ range as
this : ⨅ i, as i = x
⊢ ⋃ i, Ici (as i) = Ioi x
tactic: rw [← this] at obs
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
S : Type u_5
π : ι → Type u_6
inst✝³ : CompleteLinearOrder R
inst✝² : TopologicalSpace R
inst✝¹ : OrderTopology R
x : R
as : ι → R
x_lt : ∀ (i : ι), x < as i
F : Filter ι
inst✝ : F.NeBot
as_lim : Tendsto as F (𝓝 x)
obs : ⨅ i, as i ∉ range as
this : ⨅ i, as i = x
⊢ ⋃ i, Ici (as i) = Ioi x
state_after: ι : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
S : Type u_5
π : ι → Type u_6
inst✝³ : CompleteLinearOrder R
inst✝² : TopologicalSpace R
inst✝¹ : OrderTopology R
x : R
as : ι → R
x_lt : ∀ (i : ι), x < as i
F : Filter ι
inst✝ : F.NeBot
as_lim : Tendsto as F (𝓝 x)
obs : ⨅ i, as i ∉ range as
this : ⨅ i, as i = x
⊢ ⋃ i, Ici (as i) = Ioi (⨅ i, as i)
tactic: rw [← this]
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
S : Type u_5
π : ι → Type u_6
inst✝³ : CompleteLinearOrder R
inst✝² : TopologicalSpace R
inst✝¹ : OrderTopology R
x : R
as : ι → R
x_lt : ∀ (i : ι), x < as i
F : Filter ι
inst✝ : F.NeBot
as_lim : Tendsto as F (𝓝 x)
obs : ⨅ i, as i ∉ range as
this : ⨅ i, as i = x
⊢ ⋃ i, Ici (as i) = Ioi (⨅ i, as i)
state_after: no goals
tactic: exact iUnion_Ici_eq_Ioi_iInf obs