# Difference between revisions of "Cauchy sequence"

In Mathematics, a Cauchy sequence is an infinite sequence the members of which get progressively closer to each other. More formally, a Cauchy sequence, , in a metric space M, with distance function d, is a sequence such that for any positive real number e, there is some integer N, such that , whenever n and m are greater than N.
A metric space where all Cauchy sequences are convergent is said to be complete. For any metric space, there exists a complete metric space, containing it. Thus, in one sense every Cauchy sequence is convergent to a limit, but in a larger set than that considered, when defining the sequence. In the case of the rational numbers, , which is not complete, the larger set (called the completion) of , is the set of real numbers, . The sequence above has a limit, in the set of real numbers, but not in the set of rational numbers.