AClosedSubsetOfACompleteMetricSpaceIsComplete 2006-12-31 10:04:49 2006-12-31 10:04:49 Result complete % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $X$ be a complete metric space, and let $Y \subseteq X$ be a closed subset of $X$. Then $Y$ is complete. {\bf Proof} Let $\{ y_n \} \subseteq Y$ be a Cauchy sequence in $Y$. Then by the completeness of $X$, $y_n \rightarrow x$ for some $x \in X$. Then every neighborhood of $x$ contains points in $Y$, so $x \in \overline Y = Y$.