Theorems · Theorem · general topology

tendsto_of_le_liminf_of_limsup_le

∀ {α : Type u_2} {β : Type u_3} [inst : ConditionallyCompleteLinearOrder α] [inst_1 : TopologicalSpace α]
  [OrderTopology α] {f : Filter β} {u : β → α} {a : α},
  a ≤ Filter.liminf u f →
    Filter.limsup u f ≤ a →
      autoParam (Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f u) tendsto_of_le_liminf_of_limsup_le._auto_1 →
        autoParam (Filter.IsBoundedUnder (fun x1 x2 => x1 ≥ x2) f u) tendsto_of_le_liminf_of_limsup_le._auto_3 →
          Filter.Tendsto u f (nhds a)

If a number a is less than or equal to the liminf of a function f at some filter and is greater than or equal to the limsup of f, then f tends to a along this filter.

Defined in

Mathlib.Topology.Order.LiminfLimsup

Cited by

7 results in Mathlib

Foundations

Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ConditionallyCompleteLinearOrderTopologicalSpaceOrderTopology