Difference between revisions of "Bolzano-Weierstrass theorem"
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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 | + | 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 [[#References|[1]]]; it was later also independently deduced by K. Weierstrass. | The theorem was demonstrated by B. Bolzano [[#References|[1]]]; it was later also independently deduced by K. Weierstrass. | ||
Revision as of 21:14, 20 October 2012
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=28588
Bolzano-Weierstrass theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bolzano-Weierstrass_theorem&oldid=28588
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article