# characterization of convergence of sequences in metric spaces

Let $(M,d)$ be a metric space, and let $\mathbb{N}={P}_{1}\bigcup \mathrm{\cdots}\bigcup {P}_{k}$ be a partition of the set of natural numbers such that ${P}_{i}$ is infinite^{} for every $i$, that is, there is a bijection ${f}_{i}:\mathbb{N}\to {P}_{i}$. Then, given a sequence ${({x}_{n})}_{n\in \mathbb{N}}$, it converges^{} to $x\in M$ if and only if the subsequence

$${({x}_{{f}_{i}(n)})}_{n}$$ |

converges to $x$ for every $i=1,\mathrm{\cdots},k$.

Examples

If you have a sequence ${({x}_{n})}_{n}$ and a natural number^{} $k$, and you know that it converges to $x$ for every corresponding subsequence over the classes of remainders^{} modulo $k$, then it converges to $x$.

Title | characterization of convergence of sequences in metric spaces |
---|---|

Canonical name | CharacterizationOfConvergenceOfSequencesInMetricSpaces |

Date of creation | 2013-03-22 15:06:10 |

Last modified on | 2013-03-22 15:06:10 |

Owner | gumau (3545) |

Last modified by | gumau (3545) |

Numerical id | 4 |

Author | gumau (3545) |

Entry type | Theorem^{} |

Classification | msc 40A05 |

Classification | msc 54E35 |