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Bolzano-Weierstrass theorem

From Encyclopedia of Mathematics
Revision as of 21:18, 20 October 2012 by Boris Tsirelson (talk | contribs) ("more general", not "more-general")
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Each bounded sequence of numbers contains a convergent subsequence. The theorem applies both to real and complex numbers. It can be generalized to include more general objects, e.g. any bounded infinite set in $n$-dimensional Euclidean space has at least one limit point in that space. There exist analogues of this theorem for even more general spaces.

The theorem was demonstrated by B. Bolzano [1]; it was later also independently deduced by K. Weierstrass.

References

[1] B. Bolzano, Abhandlungen der königlichen böhmischen Gesellschaft der Wissenschaften. v. (1817)
How to Cite This Entry:
Bolzano-Weierstrass theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bolzano-Weierstrass_theorem&oldid=30994
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article