30A99-WeierstrassMtest.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{WeierstrassMtest}
\pmcreated{2013-03-22 12:56:11}
\pmmodified{2013-03-22 12:56:11}
\pmowner{yark}{2760}
\pmmodifier{yark}{2760}
\pmtitle{Weierstrass M-test}
\pmrecord{13}{33293}
\pmprivacy{1}
\pmauthor{yark}{2760}
\pmtype{Theorem}
\pmcomment{trigger rebuild}
\pmclassification{msc}{30A99}
\pmrelated{AbsoluteConvergence}

\endmetadata

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\usepackage{amsmath}
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\newcommand{\N}{\mathbb{N}}
\begin{document}
Let $X$ be any set, $\{f_n\}_{n\in\N}$ a sequence of real or complex valued functions on $X$ and $\{M_n\}_{n\in\N}$ a sequence of non-negative real numbers.
Suppose that, for each $n \in \N$ and $x \in X$, we have
$|f_n(x)| \le M_n$. Then $f=\sum_{n=1}^{\infty} f_n$ converges uniformly if
$\sum_{n=1}^{\infty} M_n$ converges.
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\end{document}