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
Does a function sequence decreasing monotonically to 0 converge uniformly?
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general-topology
functions
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0Anyhow, to improve the chances of someone finding this one, I edited the title of the question. – 2012-12-03
1 Answers
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Yes. This is known as Dini's theorem.