Cauchy sequence
In a metric space

Since the definition of a Cauchy sequence only involves metric concepts, it is straightforward to generalize it to any metric space X. To do so, the absolute value | x m − x n | {\displaystyle \left|x_{m}-x_{n}\right|} is replaced by the distance d (x m, x n) {\displaystyle d\left(x_{m},x_{n}\right)} (where d denotes a metric) between x m {\displaystyle x_{m}} and x n. {\displaystyle x_{n}.}

Formally, given a metric space (X, d), {\displaystyle (X,d),} a sequence x 1, x 2, x 3, … {\displaystyle x_{1},x_{2},x_{3},\ldots } is Cauchy, if for every positive real number ε > 0 {\displaystyle \varepsilon >0} there is a positive integer N {\displaystyle N} such that for all positive integers m, n > N, {\displaystyle m,n>N,} the distance d (x m, x n) < ε. {\displaystyle d\left(x_{m},x_{n}\right)<\varepsilon.}

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: the property of a space that every Cauchy sequence converges in the space is called completeness, and is detailed below.
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