Cauchy sequence

English

Etymology

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

Noun

1. Template:Lb Any sequence $x_n$ in a metric space with metric d such that for every $ϵ>0$ there exists a natural number N such that for all $k,m\ge N$, $d(x_k,x_m)<ϵ$.
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Usage notes

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

Derived terms

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Related terms

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Translations

• Danish: Template:T
• Finnish: Template:T
• German: Template:T, Template:T
• Italian: Template:T
• Serbo-Croatian: Template:T
• Swedish: Template:T+