Cauchy sequence

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2020 Mathematics Subject Classification: Primary: 40A05 Secondary: 54E35 [MSN][ZBL]

Cauchy sequence, of points in a metric space $(X,d)$

A sequence $\{x_i\}$ of elements in a metric space $(X,d)$ such that for any $\varepsilon > 0$ there is a number $N$ such that \[ d (x_n, x_m) < \varepsilon \qquad \forall m,n\geq N\, . \] The latter is also called Cauchy condition. A convergent sequence is always necessarily a Cauchy sequence. However the converse is not necessarily true. A metric space with the property that any Cauchy sequence has a limit is called complete, see also Cauchy criteria.

The concept of Cauchy sequence can be generalized to Cauchy nets (see also Net; Net (of sets in a topological space), Generalized sequence and Cauchy filter) in a uniform space. Let $X$ be a uniform space with uniformity $\mathcal{U}$. A net $\{x_\alpha, \alpha \in A\}$ of elements $x_\alpha \in X$ (where $A$ is a directed set) is called a Cauchy net if for every element $U\in \mathcal{U}$ there is an index $\alpha_0 \in A$ such that for all \[ (x_\alpha, x_\beta)\in U \qquad \forall \alpha, \beta \geq \alpha_0\, . \]


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[KF] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
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