# Cauchy sequence

A sequence^{} ${x}_{0},{x}_{1},{x}_{2},\mathrm{\dots}$ in a metric space $(X,d)$ is a *Cauchy sequence ^{}* if, for every real number $\u03f5>0$, there exists a natural number

^{}$N$ such that $$ whenever $n,m>N$.

Likewise, a sequence ${v}_{0},{v}_{1},{v}_{2},\mathrm{\dots}$ in a topological vector space^{} $V$ is a *Cauchy sequence* if and only if for every neighborhood $U$ of $\mathrm{\U0001d7ce}$, there exists a natural number $N$ such that ${v}_{n}-{v}_{m}\in U$ for all $n,m>N$. These two definitions are equivalent^{} when the topology^{} of $V$ is induced by a metric.

Title | Cauchy sequence |
---|---|

Canonical name | CauchySequence |

Date of creation | 2013-03-22 11:55:04 |

Last modified on | 2013-03-22 11:55:04 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 10 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 54E35 |

Classification | msc 26A03 |

Synonym | fundamental sequence |

Related topic | MetricSpace |