Cauchy sequence

A sequence $x_{0},x_{1},x_{2},\\dots$ in a metric space $(X,d)$ is a Cauchy sequence if, for every real number $\\epsilon>0$ , there exists a natural number $N$ such that $d(x_{n},x_{m})<\\epsilon$ whenever $n,m>N$ .

Likewise, a sequence $v_{0},v_{1},v_{2},\\dots$ in a topological vector space $V$ is a Cauchy sequence if and only if for every neighborhood $U$ of $\\mathbf{0}$ , there exists a natural number $N$ such that $v_{n}-v_{m}\\in U$ for all $n,m>N$ . These two definitions are equivalent when the topology of $V$ is induced by a metric.

单词	CauchySequence
释义	Cauchy sequence A sequence $x_{0},x_{1},x_{2},\\dots$ in a metric space $(X,d)$ is a Cauchy sequence if, for every real number $\\epsilon>0$ , there exists a natural number $N$ such that $d(x_{n},x_{m})<\\epsilon$ whenever $n,m>N$ . Likewise, a sequence $v_{0},v_{1},v_{2},\\dots$ in a topological vector space $V$ is a Cauchy sequence if and only if for every neighborhood $U$ of $\\mathbf{0}$ , there exists a natural number $N$ such that $v_{n}-v_{m}\\in U$ for all $n,m>N$ . These two definitions are equivalent when the topology of $V$ is induced by a metric.
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