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Definition 2024

Cauchy sequence ‎(plural Cauchy sequences)

(analysis) A sequence in a normed vector space such that the difference between any two entries can be made arbitrarily small by stipulating that the two entries be sufficiently far out in the sequence.
$\lim _{n,m\to \infty }\|x_{n}-x_{m}\|=0$
(analysis) A sequence $x_{n}$ in a metric space with metric d such that for every $\epsilon >0$ there exists a natural number N so that for every $k,m\geq N$ : $d(x_{k},x_{m})<\epsilon$ .
- 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.

sequence in a normed vector space