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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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    $f_n(x) =\frac{\sin(nx)}{n}$2012-06-11
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    Take for $f_n$ a bump of height $1/n$ and width $1$ centered at $n$.2012-06-11
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    Will any type of "bump" work?2012-06-11
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    @AdamRubinson: Yes. Draw a picture.2012-06-11
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    @Adam: And please use the @ thing.2012-06-11
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    @Adam: What if $\{\sup_{\mathbb R}|f_{n}(x)|\}$ is also converges to 0, and all $f_{n}$ are positive continuous functions? Does it change the answer?2012-06-11
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    The $\sup$ converging to $0$ is the same as uniformly converging to $0$. Modify the example from @N.S. by taking the absolute value. This satisfies your latest criteria.2012-06-11
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    did - what is the @ thing?2012-06-11

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