Theorems · Theorem · general topology

EMetric.cauchySeq_iff

∀ {α : Type u} {β : Type v} [inst : PseudoEMetricSpace α] [Nonempty β] [inst_2 : SemilatticeSup β] {u : β → α},
  CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (u m) (u n) < ε

In a pseudoemetric space, Cauchy sequences are characterized by the fact that, eventually, the pseudoedistance between its elements is arbitrarily small

Defined in

Mathlib.Topology.EMetricSpace.Basic

Cited by

1 results in Mathlib

Foundations

Depth 147 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PseudoEMetricSpaceNonemptySemilatticeSup