Cauchy sequence

Etymology
Named after French mathematician (1789–1857), who made pioneering contributions to analysis.

Noun

 * 1)  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 $$ k, m \ge N $$, $$ d(x_k, x_m) < \epsilon $$.

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

Translations

 * Danish: Cauchyfølge
 * Finnish: Cauchyn jono
 * German: Cauchy-Folge, Cauchyfolge
 * Italian: successione di Cauchy
 * Serbo-Croatian: Cauchyjev niz
 * Swedish: