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Weierstrass' criterion of uniform convergence

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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
\begin{document}
\begin{thmplain}
\, \,Let the real functions $f_1(x)$, $f_2(x)$, ... be defined in the interval $[a, b]$. \,If they all \PMlinkescapetext{satisfy} the condition 
         $$|f_n(x)| \leqq M_n \quad \forall\,x\in[a, b],$$
with $\sum_{n = 1}^{\infty}M_n$ a convergent series of \PMlinkescapetext{constant terms}, then the function series 
                      $$f_1(x)\!+\!f_2(x)\!+\!\cdots$$
\PMlinkname{converges uniformly}{SumFunctionOfSeries} on the interval $[a, b]$.
\end{thmplain}

The theorem is valid also for the series with complex function terms, when one replaces the interval with a subset of $\mathbb{C}$.
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