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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016880/b0168801.png" />-dimensional Euclidean space has at least one limit point in that space. There exist analogues of this theorem for even more-general spaces.
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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.
  
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The theorem was demonstrated by B. Bolzano [[#References|[1]]]; 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====
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B. Bolzano,  ''Abhandlungen der königlichen böhmischen Gesellschaft der Wissenschaften. v.'' (1817)</TD></TR></table>
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|valign="top"|{{Ref|Bo}}|| B. Bolzano,  ''Abhandlungen der königlichen böhmischen Gesellschaft der Wissenschaften. v.''
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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=28587
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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