I need to apply the Arzela-Ascoli theorem for a sequence $f_n$. I already have the uniform bound and now it says $\frac{d}{dx} f_n(x)$ is bounded in $L^2([a,b])$ and I assume this yields the equicontinuity.
The continuity obviously follows from the integrability, but i have no idea of how to show that the sequence is indeed equicontinuous. Does anybody have some hint for me? Thank you in advance.
$|f_n(x)-f_n(y)| =|\int_x^{y}f_n'(t)\, dt|\leq \sqrt {|x-y|} \sqrt {\int_x^{y} |f_n'(t)|^{2} \, dt} \leq M\sqrt {|x-y|}$ where $M=\sqrt {\sup_n \int_a^{b} |f_n'(t)|^{2} \, dt}$ (where we have used Holder's inequality). This gives equicontinuity.