Cauchy sequence

== English == === Etymology === Named after French mathematician Augustin-Louis Cauchy (1789–1857), who made pioneering contributions to analysis. === Noun === Cauchy sequence (plural Cauchy sequences) (mathematical analysis) Any sequence x n {\displaystyle x_{n}} in a metric space with metric d such that for every ϵ > 0 {\displaystyle \epsilon >0} there exists a natural number N such that for all k , m ≥ N {\displaystyle k,m\geq N} , d ( x k , x m ) < ϵ {\displaystyle d(x_{k},x_{m})<\epsilon } . ==== 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. ==== Derived terms ==== Cauchy (adjective) ==== Related terms ==== Cauchy convergence Cauchy filter Cauchy net Cauchy space ==== Translations ====