Definition:Cauchy Sequence/Metric Space

Definition

Let $M = \struct {A, d}$ be a metric space.

Let $\sequence {x_n}$ be a sequence in $M$.

Then $\sequence {x_n}$ is a Cauchy sequence if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \map d {x_n, x_m} < \epsilon$

Also see

• Convergent Sequence in Metric Space is Cauchy Sequence

• Definition:Complete Metric Space: a metric space in which the converse holds, that is, a Cauchy sequence is convergent.

Source of Name

This entry was named for Augustin Louis Cauchy.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $7.2$: Sequential compactness: Definition $7.2.3$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces
• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $4$: Complete Normed Spaces