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Difference between revisions of "Divergent sequence"

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A sequence of points in a topological space without a limit. Every divergent sequence in a compact metric space contains a convergent subsequence. In the class of divergent sequences in a normed space one can find infinitely large sequences, i.e. sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033620/d0336201.png" /> of points such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033620/d0336202.png" />. The concept of a divergent sequence can be generalized to multiple sequences and to sequences in directed (partially ordered) sets.
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A sequence of points in a topological space without a limit. Every divergent sequence in a compact metric space contains a convergent subsequence. In the class of divergent sequences in a normed space one can find infinitely large sequences, i.e. sequences $\{x_n\}$ of points such that $\lim\limits_{n\to\infty}\|x_n\|=\infty.$ be generalized to multiple sequences and to sequences in directed (partially ordered) sets.

Revision as of 06:53, 23 January 2013

A sequence of points in a topological space without a limit. Every divergent sequence in a compact metric space contains a convergent subsequence. In the class of divergent sequences in a normed space one can find infinitely large sequences, i.e. sequences $\{x_n\}$ of points such that $\lim\limits_{n\to\infty}\|x_n\|=\infty.$ be generalized to multiple sequences and to sequences in directed (partially ordered) sets.

How to Cite This Entry:
Divergent sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divergent_sequence&oldid=14486
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article