Bolzano-Weierstrass theorem BolzanoWeierstrassTheorem 2002-02-18 19:43:40 2007-07-01 16:34:10 Theorem \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} Given any bounded sequence $(a_n)$ of real numbers, there exists a convergent subsequence $(a_{n_j})$. More generally, any sequence $(a_n)$ in a compact subset of a metric space has a convergent subsequence.