Theorems · Theorem · order theory

Filter.eventually_lt_add_pos_of_limsup_le

∀ {α : Type u_1} {β : Type u_2} [inst : ConditionallyCompleteLinearOrder α] [inst_1 : Preorder β]
  [inst_2 : AddZeroClass α] [AddLeftStrictMono α] {x ε : α} {u : β → α},
  Filter.IsBoundedUnder LE.le Filter.atTop u →
    Filter.limsup u Filter.atTop ≤ x → 0 < ε → ∀ᶠ (b : β) in Filter.atTop, u b < x + ε

If Filter.limsup u atTop ≤ x, then for all ε > 0, eventually we have u b < x + ε.

Defined in

Mathlib.Order.LiminfLimsup

Cited by

1 results in Mathlib

Foundations

Depth 18 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ConditionallyCompleteLinearOrderPreorderAddZeroClassAddLeftStrictMono