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.
Can Arzela-Ascoli theorem holds without the sequence being uniformly bounded?
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0@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