Hölder Condition and equicontinuity

Question

My question is the following: If $|f_n(x)-f_n(y)|≤M|x-y|^\alpha$ for some fixed M and $\alpha>0$ and all x,y in a compact interval, show that ${f_n}$ is uniformly equicontinuous.

As a criterion for equicontintinuity, we check by the mean value theorem, whether $|f_k(x)-f_k(y)|≤M|x-y|$. When $|x-y|≤1$, we have $|f_n(x)-f_n(y)|≤M|x-y|^\alpha≤M|x-y|$. I do not see what I can do when $|x-y|>1$, since then $M|x-y|<M|x-y|^\alpha$, and so $\frac{|f_n(x)-f_n(y)|}{|x-y|^{\alpha-1}}≤M|x-y|$. Is there a general proof structure with these problems?

Thanks!

Answer 1

$|f_n(x)-f_n(y)| <\epsilon$ whenever $|x-y|<\delta$ where $\delta =\epsilon ^{1/\alpha}$. That is all there is to it! You are trying to prove the stronger condition that the sequence is Lipschitz. In fact, Lipschitz condition may not be satisfied.