Definition:Cauchy Sequence/Metric Space
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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
- 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