Theorems · Theorem · general topology

tendsto_of_no_upcrossings

∀ {α : Type u_2} {β : Type u_3} [inst : ConditionallyCompleteLinearOrder α] [inst_1 : TopologicalSpace α]
  [OrderTopology α] [DenselyOrdered α] {f : Filter β} {u : β → α} {s : Set α},
  Dense s →
    (∀ a ∈ s, ∀ b ∈ s, a < b → ¬((∃ᶠ (n : β) in f, u n < a) ∧ ∃ᶠ (n : β) in f, b < u n)) →
      autoParam (Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f u) tendsto_of_no_upcrossings._auto_1 →
        autoParam (Filter.IsBoundedUnder (fun x1 x2 => x1 ≥ x2) f u) tendsto_of_no_upcrossings._auto_3 →
          ∃ c, Filter.Tendsto u f (nhds c)

Assume that, for any a < b, a sequence cannot be infinitely many times below a and above b. If it is also ultimately bounded above and below, then it has to converge. This even works if a and b are restricted to a dense subset.

Defined in

Mathlib.Topology.Order.LiminfLimsup

Cited by

2 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ConditionallyCompleteLinearOrderTopologicalSpaceOrderTopologyDenselyOrdered