# Cauchy sequence

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### Etymology

Named after French mathematician Augustin-Louis Cauchy (1789–1857), who made pioneering contributions to analysis.

### Noun

Cauchy sequence (plural Cauchy sequences)

1. (mathematical analysis) Any sequence $x_{n}$ in a metric space with metric d such that for every $\epsilon >0$ there exists a natural number N such that for all $k,m\geq N$ , $d(x_{k},x_{m})<\epsilon$ .
• 1955, [Van Nostrand], John L. Kelley, General Topology, 1975, Springer, page 174,
However, it is possible to derive topological results from statements about Cauchy sequences; for example, a subset A of the space of real numbers is closed if and only if each Cauchy sequence in A converges to some point of A.
• 2000, George Bachman, Lawrence Narici, Functional Analysis, page 52,
In the case of the real line, every Cauchy sequence converges; that is, being a Cauchy sequence is sufficient to guarantee the existence of a limit. In the general case, however, this is not so. If a metric space does have the property that every Cauchy sequence converges, the space is called a complete metric space.
• 2012, David Applebaum, Limits, Limits Everywhere: The Tools of Mathematical Analysis, Oxford University Press, page 153,
Cantor first redefined Cauchy sequences using rational numbers only. [] Cantor's idea was to define the real number line as the collection of all (rational) Cauchy sequences.

#### Usage notes

The formal definition of Cauchy sequence represents a formulation of the notion of convergence without reference to a supposed element to which the sequence converges. In fact, the spaces of most interest to analysis are those, called complete, in which such limits do exist within the space.