Editing Limit of a 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 <math>x_n</math> versus ''n''. Visually, we see that the sequence appears to be converging to a limit point as the terms in the sequence become closer together as ''n'' increases. In the [[real numbers]] every Cauchy sequence converges to some limit.]]

A Cauchy sequence is a sequence whose terms ultimately become arbitrarily close together, after sufficiently many initial terms have been discarded. 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 the ''Cauchy criterion for convergence of sequences'': a sequence of real numbers is convergent if and only if it is a Cauchy sequence. This remains true in other [[complete metric space]]s.