# Cauchy Sequence - in A Metric Space

In A Metric Space

To define Cauchy sequences in any metric space X, the absolute value is replaced by the distance (where d : X × XR with some specific properties, see Metric (mathematics)) between and .

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 natural numbers 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.

### Other articles related to "in a metric space":

Neighbourhood (mathematics) - In A Metric Space
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### Famous quotes containing the word space:

At first thy little being came:
If nothing once, you nothing lose,
For when you die you are the same;
The space between, is but an hour,
The frail duration of a flower.
Philip Freneau (1752–1832)