Difference between revisions of "Cauchy sequence"

From Citizendium
Jump to navigation Jump to search
imported>Hendra I. Nurdin
(Stub for Cauchy sequence)
 
imported>Hendra I. Nurdin
(oops, forgot to mention for all n,m>N(\epsilon))
Line 2: Line 2:


==Formal definition==
==Formal definition==
Let <math>(X,d)</math> be a metric space. Then a sequence <math>x_1,x_2,\ldots</math> of elements in ''X'' is a Cauchy sequence if for any real number <math>\epsilon>0</math> there exists a positive integer <math>N(\epsilon)</math>, dependent on <math>\epsilon</math>, such that <math>d(x_n,x_m)<\epsilon</math>. In [[limit]] notation this is written as <math>\mathop{\lim}_{n,m \rightarrow \infty}d(x_m,x_n)=0</math>.
Let <math>(X,d)</math> be a metric space. Then a sequence <math>x_1,x_2,\ldots</math> of elements in ''X'' is a Cauchy sequence if for any real number <math>\epsilon>0</math> there exists a positive integer <math>N(\epsilon)</math>, dependent on <math>\epsilon</math>, such that <math>d(x_n,x_m)<\epsilon</math> for all <math>m,n>N(\epsilon)</math>. In [[limit]] notation this is written as <math>\mathop{\lim}_{n,m \rightarrow \infty}d(x_m,x_n)=0</math>.


[[Category:Mathematics_Workgroup]]
[[Category:Mathematics_Workgroup]]
[[Category:CZ Live]]
[[Category:CZ Live]]

Revision as of 20:02, 2 October 2007

In mathematics, a Cauchy sequence is sequence in a metric space with the property that elements in that sequence cluster together more and more as the sequence progresses. Another way of thinking of the clustering is that the distance between any two elements diminishes as their indexes grow larger and larger.

Formal definition

Let be a metric space. Then a sequence of elements in X is a Cauchy sequence if for any real number there exists a positive integer , dependent on , such that for all . In limit notation this is written as .