Cauchy sequence

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

Cauchy sequence (plural Cauchy sequences)

1. (mathematical 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.

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.

• Cauchy convergence
• Cauchy filter
• Cauchy net
• Cauchy space