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In mathematical analysis, a Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses. They are named after the French mathematician Augustin Louis Cauchy.
Formally, a sequence x_{1}, x_{2}, x_{3}, ... in a metric space (M, d) is called a Cauchy sequence (or Cauchy for short) if for every positive real number r, there is an integer N such that for all integers m and n greater than N the distance d(x_{m}, x_{n}) is less than r. 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 M. Nonetheless, this does not need to be the case.
A metric space in which every Cauchy sequence has a limit is called complete. The real numbers are complete, and the standard construction of the real numbers involves Cauchy sequences of rational numbers. The rational numbers themselves are not complete: a sequence of rational numbers can have the square root of two as its limit, for example. See Complete space for an example of a Cauchy sequences of rational numbers that does not have a rational limit.
Every convergent sequence is a Cauchy sequence, and every Cauchy sequence is bounded. If f : M > N is a uniformly continuous map between the metric spaces M and N and (x_{n}) is a Cauchy sequence in M , then (f(x_{n})) is a Cauchy sequence in N. If (x_{n}) and (y_{n}) are two Cauchy sequences in the rational, real or complex numbers, then the sum (x_{n} + y_{n}) and the product (x_{n}y_{n}) are also Cauchy sequences.
A net (x_{α}) in a uniform space X is a Cauchy net if for every entourage V there exists an α_{0} such that for all α, β > α_{0} we have (x_{α}, x_{β}) in V.
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