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{{MSC|40A05|54E35}}
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''Cauchy sequence, of points in a metric space $(X,d)$''

A  sequence $\{x_i\}$ of elements in a [[Metric space|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 metric space|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|uniform space]]. Let $X$ be a uniform space  with uniformity $\mathcal{U}$. A net $\{x_\alpha, \alpha \in A\}$ (where $A$ is a [[Directed set|directed set]] of elements $x_\alpha \in X$, 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 
\[
(x_\alpha, x_\beta)\in U \qquad \forall \alpha, \beta \geq \alpha_0\, .
\]

====References====
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|valign="top"|{{Ref|Al}}|| P.S.  Aleksandrov,   "Einführung in die Mengenlehre und die allgemeine   Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from   Russian)
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|valign="top"|{{Ref|Du}}|| J. Dugundji,   "Topology" , Allyn &amp; Bacon  (1966)  {{MR|0193606}} {{ZBL|0144.21501}}
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|valign="top"|{{Ref|Ke}}|| J.L. Kelley,    "General topology" , Springer   (1975) 
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|valign="top"|{{Ref|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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