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.

accumulation point | Bolzano's theorem | Cantor's intersection theorem | Heine-Borel theorem | intermediate value theorem