**In A Metric Space**

To define Cauchy sequences in any metric space X, the absolute value is replaced by the *distance* (where *d* : *X* × *X* → **R** 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*.

Read more about this topic: Cauchy Sequence

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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)