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.

The theorem was demonstrated by B. Bolzano [Bo]; it was later also independently deduced by K. Weierstrass.


[Bo] B. Bolzano, Abhandlungen der königlichen böhmischen Gesellschaft der Wissenschaften. v.
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Bolzano-Weierstrass theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article