Theorems · Theorem · general topology

Metric.cauchySeq_iff

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

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

Defined in

Mathlib.Topology.MetricSpace.Cauchy

Cited by

4 results in Mathlib

Foundations

Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PseudoMetricSpaceNonemptySemilatticeSup