Difference between revisions of "Bolzano-Weierstrass theorem"

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(a start; need to improve and help is welcome)
 
(Put in stuff about accumulation points. I hope I haven't messed up the economics stuff, but it doesn't seem very clear, either before or after. Can someone tell us just what the theorem says about e)
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The Bolzano-Weierstrass theorem is a statement about the convergence of sequences, and is a basic theorem of [[real analysis]] in [[mathematics]]. 
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The Bolzano-Weierstrass theorem holds that if a sequence of real numbers is finite with endpoints, then it must have a subsequence that converges. The theorem holds that an equilibrium exists in the form of a convergent subsequence.  The proof of the existence of such an equilibrium has become useful to [[economics]].
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The '''Bolzano-Weierstrass theorem''' is an extremely important theorem of [[real analysis]] in [[mathematics]].  It states that any infinite, bounded set of [[real numbers]] has an [[accumulation point]].  Equivalently, it states that any bounded infinite sequence of real numbers has a subsequence that converges.
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The theorem has useful applications to [[economics]].  It asserts the existence of an equilibrium in certain economic problems.
  
 
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Revision as of 03:08, November 7, 2009

This article/section deals with mathematical concepts appropriate for late high school or early college.

The Bolzano-Weierstrass theorem is an extremely important theorem of real analysis in mathematics. It states that any infinite, bounded set of real numbers has an accumulation point. Equivalently, it states that any bounded infinite sequence of real numbers has a subsequence that converges.

The theorem has useful applications to economics. It asserts the existence of an equilibrium in certain economic problems.

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