 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 positive integers "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"