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Difference between revisions of "Cauchy sequence"

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The same as a [[Fundamental sequence|fundamental sequence]].
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{{MSC|40A05|54E35}}
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''Cauchy sequence, of points in a metric space $(X,d)$''
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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
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\[
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d (x_n, x_m) < \varepsilon \qquad \forall m,n\geq N\, .
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\]
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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]].
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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 $\{x_\alpha, \alpha \in A\}$ of elements $x_\alpha \in X$ (where $A$ is a [[Directed set|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
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\[
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(x_\alpha, x_\beta)\in U \qquad \forall \alpha, \beta \geq \alpha_0\, .
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\]
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====References====
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{|
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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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Latest revision as of 10:12, 9 December 2013

2010 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 \[ (x_\alpha, x_\beta)\in U \qquad \forall \alpha, \beta \geq \alpha_0\, . \]

References

[Al] P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[Du] J. Dugundji, "Topology" , Allyn & Bacon (1966) MR0193606 Zbl 0144.21501
[Ke] J.L. Kelley, "General topology" , Springer (1975)
[KF] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
How to Cite This Entry:
Cauchy sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_sequence&oldid=13349