Theorems · Theorem · general topology

UniformCauchySeqOn.cauchy_map

∀ {α : Type u_1} {β : Type u_2} {ι : Type u_4} [inst : UniformSpace β] {F : ι → α → β} {s : Set α} {x : α}
  {p : Filter ι} [hp : p.NeBot], UniformCauchySeqOn F p s → x ∈ s → Cauchy (Filter.map (fun i => F i x) p)

If a sequence of functions is uniformly Cauchy on a set, then the values at each point form a Cauchy sequence.

Defined in

Mathlib.Topology.UniformSpace.UniformConvergence

Cited by

2 results in Mathlib

Foundations

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

Assumes

UniformSpaceFilter.NeBot