# a closed subset of a complete metric space is complete

Let $X$ be a complete metric space, and let $Y\subseteq X$ be a closed subset of $X$. Then $Y$ is complete^{}.

Proof

Let $\{{y}_{n}\}\subseteq Y$ be a Cauchy sequence^{} in $Y$. Then by the completeness of $X$, ${y}_{n}\to x$ for some $x\in X$. Then every neighborhood^{} of $x$ contains points in $Y$, so $x\in \overline{Y}=Y$.

Title | a closed subset of a complete metric space is complete |
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Canonical name | AClosedSubsetOfACompleteMetricSpaceIsComplete |

Date of creation | 2013-03-22 16:31:26 |

Last modified on | 2013-03-22 16:31:26 |

Owner | ehremo (15714) |

Last modified by | ehremo (15714) |

Numerical id | 4 |

Author | ehremo (15714) |

Entry type | Result |

Classification | msc 54E50 |