# Cauchy sequence

## English

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### Etymology

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

Historically, Cauchy sequences were initially imagined in the context of real numbers (where the metric is the absolute value of the difference between two numbers). Every real number is in principle definable as the limit of a Cauchy sequence of rational numbers (as is implicit in the decimal representation of real numbers), and every such sequence converges to a real number.

The idea of using Cauchy sequences to define real numbers is attributed to Georg Cantor (1845–1918).

### Noun

Cauchy sequence (plural Cauchy sequences)

1. (analysis) Any sequence ${\displaystyle x_{n}}$ in a metric space with metric d such that for every ${\displaystyle \epsilon >0}$ there exists a natural number N such that for all ${\displaystyle k,m\geq N}$, ${\displaystyle d(x_{k},x_{m})<\epsilon }$.
• 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.