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If $\{f_{n}(x)\}$ is a sequence of continuous functions on $\mathbb R$, with $|f_{n}(x)|\leq C, \forall n$, and $\lim_{n\to\infty}f_{n}(x)=0$ uniformly on $\mathbb R$, does there exist a subsequence of $\{f_{n}\}$ which is decreasing on $\mathbb R$?

Edit: What if $\{\sup_{\mathbb R}|f_{n}(x)|\}$ is also converges to 0, and all $f_{n}$ are positive continuous functions with the above properties?

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    did - what is the @ thing?2012-06-11

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No. Consider $f_n(x) \equiv -1/n$. Then $|f_n(x)|\le 1 \ \forall n $, $f_n \rightarrow 0$ uniformily on $\mathbb{R}$ and All subsequence of $(f_n)$ is strictly increasing.

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    oooh yeah I read it as 1/x. My bad2012-06-11