Editing Sequence (section)

===Cauchy sequences===
{{Main|Cauchy sequence}}

[[File:Cauchy sequence illustration.svg|350px|thumb| The plot of a Cauchy sequence (''X<sub>n</sub>''), shown in blue, as ''X<sub>n</sub>'' versus ''n''. In the graph the sequence appears to be converging to a limit as the distance between consecutive terms in the sequence gets smaller as ''n'' increases. In the [[real number]]s every Cauchy sequence converges to some limit.]]

A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion of a Cauchy sequence is important in the study of sequences in [[metric spaces]], and, in particular, in [[real analysis]]. One particularly important result in real analysis is ''Cauchy characterization of convergence for sequences'':
:A sequence of real numbers is convergent (in the reals) if and only if it is Cauchy.
In contrast, there are Cauchy sequences of [[rational numbers]] that are not convergent in the rationals, e.g. the sequence defined by <math>x_1 = 1</math> and <math>x_{n+1} = \tfrac12\bigl(x_n + \tfrac{2}{x_n}\bigr)</math> is Cauchy, but has no rational limit (cf. {{slink|Cauchy sequence#Non-example: rational numbers}}). More generally, any sequence of rational numbers that converges to an [[irrational number]] is Cauchy, but not convergent when interpreted as a sequence in the set of rational numbers.

Metric spaces that satisfy the Cauchy characterization of convergence for sequences are called [[complete metric space]]s and are particularly nice for analysis.