uniform convergence UniformConvergence 2002-12-09 06:02:22 2005-07-02 11:43:08 Definition uniformly convergent % 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 any set, and let $(Y,d)$ be a metric space. A sequence $f_1,f_2,\dots$ of functions mapping $X$ to $Y$ is said to be \emph{uniformly convergent} to another function $f$ if, for each $\varepsilon>0$, there exists $N$ such that, for all $x$ and all $n>N$, we have $d(f_n(x),f(x))<\varepsilon$. This is denoted by $f_n\xrightarrow[]{u} f$, or ``$f_n\rightarrow f$ uniformly'' or, less frequently, by $f_n\rightrightarrows f$.