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

Also see

 * Convergent Sequence in Metric Space is Cauchy Sequence


 * Definition:Complete Metric Space: a metric space in which the converse holds, i.e. a Cauchy sequence is convergent.