Difference between revisions of "Bolzano-Weierstrass theorem"
From Encyclopedia of Mathematics
m (TeX) |
("more general", not "more-general") |
||
| Line 1: | Line 1: | ||
| − | 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 | + | 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 21:18, 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=28589
Bolzano-Weierstrass theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bolzano-Weierstrass_theorem&oldid=28589
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article