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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“It is the space inside that gives the drum its sound.”
—Hawaiian saying no. 1189, ‘lelo No’Eau, collected, translated, and annotated by Mary Kawena Pukui, Bishop Museum Press, Hawaii (1983)