Editing Cauchy sequence (section)

==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 <math>\left|x_m - x_n\right|</math> is replaced by the distance <math>d\left(x_m, x_n\right)</math> (where ''d'' denotes a [[Metric (mathematics)|metric]]) between <math>x_m</math> and <math>x_n.</math>

Formally, given a [[metric space]] <math>(X, d),</math> a sequence of elements of <math>X</math>
<math display="block">x_1, x_2, x_3, \ldots</math>
is Cauchy, if for every positive [[real number]] <math>\varepsilon > 0</math> there is a positive [[integer]] <math>N</math> such that for all positive integers <math>m, n > N,</math> the distance
<math display="block">d\left(x_m, x_n\right) < \varepsilon.</math>

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 of a sequence|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.