In Metric Space

Let be a sequence of points in the metric space . We say that this sequence is a Cauchy sequence iff for every , there exists an such that for all .

Convergent Sequences Are Cauchy Sequences

Let be a sequence which converges to some limit , then is also a Cauchy sequence.

The opposite is not true.

In certain spaces, Cauchy sequences do not necessarily converge. For example, consider the Cauchy sequence in While the sequence is convergent in , it does not converge in .

A metric space where all Cauchy sequences converges is called a complete space.

Relationship with Subsequence and Convergent

Let be a Cauchy sequence. Suppose that there is some subsequence of this sequence which converges to a limit , then the original sequence also converges to .