# 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^{} |
---|---|

Canonical name | BolzanoWeierstrassTheorem |

Date of creation | 2013-03-22 12:22:21 |

Last modified on | 2013-03-22 12:22:21 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 10 |

Author | yark (2760) |

Entry type | Theorem |

Classification | msc 26A06 |

Classification | msc 40A05 |

Synonym | Bolzano-Weierstraß theorem |

Related topic | ConvergentSequence |

Related topic | SequentiallyCompact |

Related topic | AlternateStatementOfBolzanoWeierstrassTheorem |