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 .