# Bolzano-Weierstrass theorem

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.

Title Bolzano-Weierstrass theorem BolzanoWeierstrassTheorem 2013-03-22 12:22:21 2013-03-22 12:22:21 yark (2760) yark (2760) 10 yark (2760) Theorem msc 26A06 msc 40A05 Bolzano-Weierstraß theorem ConvergentSequence SequentiallyCompact AlternateStatementOfBolzanoWeierstrassTheorem