54A20-ProofOfDinisTheorem.tex

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\pmtitle{proof of Dini's theorem}
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\begin{document}
Without loss of generality we will assume that $X$ is compact and, by replacing
$f_n$ with $f-f_n$, that the net converges monotonically to 0. 

Let $\epsilon > 0$.
For each $x\in X$, we can choose an $n_x$, such that $f_{n_x}(x) <
\epsilon/2$. Since $f_{n_x}$ is continuous, 
there is an open
neighbourhood $U_x$ of $x$, such that for each $y\in U_x$, we have $f_{n_x}(y)
< \epsilon/2$. The open sets $U_x$ cover $X$, which is compact, so we can choose
finitely many $x_1, \ldots, x_k$ such that the $U_{x_i}$ also cover $X$. Then,
if $N\geq n_{x_1}, \ldots, n_{x_k}$, we have $f_n(x) < \epsilon$ for each
$n\geq N$ and $x\in X$, since the sequence $f_n$ is monotonically decreasing.
Thus, $\{f_n\}$ converges to 0 uniformly on $X$, which was to be proven.
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