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Difference between revisions of "Cauchy sequence"

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''Cauchy sequence, of points in a metric space $(X,d)$''
 
''Cauchy sequence, of points in a metric space $(X,d)$''
  
A  sequence $\{p_i\}$ of elements in a [[Metric space|metric space]] $(X,d)$ such that for any $\varepsilon > 0$ there is a number $N$ such that
+
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_, x_m) < \varepsilon \qquad \forall m,n\geq N\, .
+
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 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]])

Revision as of 10:11, 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 $\{x_\alpha, \alpha \in A\}$ (where $A$ is a 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

[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=30874