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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! – checkmath Mar 6 '12 at 3:35
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 $x_1$, $x_2$. – Harald Hanche-Olsen Mar 6 '12 at 6:23
I am sorry for losing a assumption which is that there exists a constant $M>0$ independent of $n$ such that $\min_{x\in[-n,n]}|f_n(x)|<M$. Then the limit function $f(x)$ will not be constant function, since $|f(x)|\longrightarrow+\infty$ as $|x|\longrightarrow\infty$. – Dong Wu Mar 12 '12 at 8:26
@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. – Stefan Smith Oct 30 '13 at 23:31

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