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 .
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 (17521832)