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Bolzano-Weierstrass Theorem


Every bounded infinite set in R^n has an accumulation point.

For n=1, an infinite subset of a closed bounded set S has an accumulation point in S. For instance, given a bounded sequence a_n, with -C<=a_n<=C for all n, it must have a monotonic subsequence a_(n_k). The subsequence a_(n_k) must converge because it is monotonic and bounded. Because S is closed, it contains the limit of a_(n_k).

The Bolzano-Weierstrass theorem is closely related to the Heine-Borel theorem and Cantor's intersection theorem, each of which can be easily derived from either of the other two.


See also

Accumulation Point, Bolzano's Theorem, Cantor's Intersection Theorem, Heine-Borel Theorem, Intermediate Value Theorem

Portions of this entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd and Weisstein, Eric W. "Bolzano-Weierstrass Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Bolzano-WeierstrassTheorem.html

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