Definition:Cauchy Sequence/Metric Space

Definition
Let $M = \left({A, d}\right)$ be a metric space.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $M$.

Then $\left \langle {x_n} \right \rangle$ is a Cauchy sequence iff:
 * $\forall \epsilon \in \R: \epsilon > 0: \exists N: \forall m, n \in \N: m, n \ge N: d \left({x_n, x_m}\right) < \epsilon$

Also see

 * Convergent Sequence is Cauchy Sequence


 * A complete metric space is defined as being a metric space in which the converse holds, i.e. a Cauchy sequence is convergent.