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Suppose $\{f_n\}$ be a sequence of continuous function$f_n:S\to \mathbb{R}$ where $S\subset \mathbb{R}$ and $S$ is compact. Suppose for $\{f_n(x)\}$ monotonic decreasing to zero for any $x\in S$. Is $\{f_n\}$ uniformly converge to $ 0$? I know all the definition of convergence and uniformly convergence and compact but still not sure how to start or prove it

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See also [$f_1,f_2,\dots$ continuous on $[0,1]$ s.t $f_1 \geq f_2\geq \cdots$ and $\lim_{n\to\infty}f_n(x)=0$.](math.stackexchange.com/q/121603) – user51627 Dec 3 '12 at 17:30
Hmm. I didn't see that when I searched for Dini's theorem prior to answering. Though I now realize that I would have gotten more relevant hits if I had used quotes. – Harald Hanche-Olsen Dec 3 '12 at 17:33
Anyhow, to improve the chances of someone finding this one, I edited the title of the question. – Harald Hanche-Olsen Dec 3 '12 at 17:37

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Yes. This is known as Dini's theorem.

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