Cauchy sequence

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Noun

Cauchy sequence ‎(plural Cauchy sequences)

1. (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.
${\displaystyle \lim _{n,m\to \infty }\|x_{n}-x_{m}\|=0}$
2. (analysis) A sequence ${\displaystyle x_{n}}$ in a metric space with metric d such that for every ${\displaystyle \epsilon >0}$ there exists a natural number N so that for every ${\displaystyle k,m\geq N}$ : ${\displaystyle d(x_{k},x_{m})<\epsilon }$.
• 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.