Difference between revisions of "Divergent sequence"
From Encyclopedia of Mathematics
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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 $\lim\limits_{n\to\infty}\|x_n\|=\infty$. The concept of a divergent sequence can be generalized to multiple sequences and to sequences in directed (partially ordered) sets. | 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 $\lim\limits_{n\to\infty}\|x_n\|=\infty$. The concept of a divergent sequence can be generalized to multiple sequences and to sequences in directed (partially ordered) sets. | ||
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| + | See also: [[Convergence, types of]]. | ||
Latest revision as of 20:54, 17 December 2016
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 $\lim\limits_{n\to\infty}\|x_n\|=\infty$. The concept of a divergent sequence can be generalized to multiple sequences and to sequences in directed (partially ordered) sets.
See also: Convergence, types of.
How to Cite This Entry:
Divergent sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divergent_sequence&oldid=40037
Divergent sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divergent_sequence&oldid=40037
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article