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Difference between revisions of "Bolzano-Weierstrass theorem"

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Each bounded sequence of real (or complex) numbers contains a convergent subsequence. 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.
 
Each bounded sequence of real (or complex) numbers contains a convergent subsequence. 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. Bolzaon {{Cite|Bo}}; it was later also independently deduced by K. Weierstrass.
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The theorem was demonstrated by B. Bolzano {{Cite|Bo}}; it was later also independently deduced by K. Weierstrass.
  
 
====References====
 
====References====

Latest revision as of 08:01, 2 May 2014

Each bounded sequence of real (or complex) numbers contains a convergent subsequence. 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 [Bo]; it was later also independently deduced by K. Weierstrass.

References

[Bo] B. Bolzano, Abhandlungen der königlichen böhmischen Gesellschaft der Wissenschaften. v.
How to Cite This Entry:
Bolzano-Weierstrass theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bolzano-Weierstrass_theorem&oldid=32120
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article