Every bounded infinite set in has an accumulation point.
For , an infinite subset of a closed bounded set has an accumulation point in . For instance, given a bounded sequence , with for all , it must have a monotonic subsequence . The subsequence must converge because it is monotonic and bounded. Because is closed, it contains the limit of .
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.