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I have another question on Arzela-Ascoli theorem. That is if the sequence $f_n(x)$ is defined in [-n,n] and $f_{n}^{'}(x)\rightarrow0$ uniformly in $R$, can I use the Arzela-Ascoli theorem? Furthermore, I need $|f(x)|\rightarrow+\infty$ as $|x|\rightarrow+\infty$, where $f(x)$ is the limit function.

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    Well Arzela-Ascoli states that you have a uniformly convergent subsequence, but your sequence is already uniformly convergent! I could not understand your point!2012-03-06
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    All you can say with your stated assumptions is that some subsequence will converge to a constant function, with the constant possibly being $\pm\infty$. The main point being that having the derivative going to 0 uniformly implies $f_n(x_1)-f_n(x_2)\to0$ for any $x_1$, $x_2$.2012-03-06
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    I am sorry for losing a assumption which is that there exists a constant $M>0$ independent of $n$ such that $\min_{x\in[-n,n]}|f_n(x)|. Then the limit function $f(x)$ will not be constant function, since $|f(x)|\longrightarrow+\infty$ as $|x|\longrightarrow\infty$.2012-03-12
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    @DongWu : If you just let $f_n(x) = n$ for all $n$, it seems to satisfy all your assumptions, and it does not converge in any way. I think you need more assumptions. You also seem to be assuming there is a limit function, when the whole point of the Arzela-Ascoli theorem is to establish the existence of a limit function. Please fix your question. Fix the _question_ itself, don't just put any corrections in a comment.2013-10-30

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