Cauchy Sequence - in A Metric Space

In A Metric Space

To define Cauchy sequences in any metric space X, the absolute value is replaced by the distance (where d : X × X → R with some specific properties, see Metric (mathematics)) between and .

Formally, given a metric space (X, d), a sequence

is Cauchy, if for every positive real number ε > 0 there is a positive integer N such that for all natural numbers m,n > N, the distance

Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ought to have a limit in X. Nonetheless, such a limit does not always exist within X.