# convergent sequence

A sequence ${x}_{0},{x}_{1},{x}_{2},\mathrm{\dots}$ in a metric space $(X,d)$ is a *convergent sequence* if there exists a point $x\in X$ such that, for every real number $\u03f5>0$, there exists a natural number^{} $N$ such that $$ for all $n>N$.

The point $x$, if it exists, is unique, and is called the *limit point* or *limit* of the sequence. One can also say that the sequence ${x}_{0},{x}_{1},{x}_{2},\mathrm{\dots}$ *converges* to $x$.

A sequence is said to be *divergent* if it does not converge.

Title | convergent sequence |

Canonical name | ConvergentSequence |

Date of creation | 2013-03-22 11:55:07 |

Last modified on | 2013-03-22 11:55:07 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 10 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 54E35 |

Classification | msc 40A05 |

Related topic | AxiomOfAnalysis |

Related topic | BolzanoWeierstrassTheorem |

Related topic | Sequence |

Defines | limit point |

Defines | limit |

Defines | converge |

Defines | diverge |

Defines | divergent sequence |