Let $\{f_n\} \subset C(-2,2)$ with $||f_n||_{L^\infty(-2,2)}\leq 1$. Assume for any $x,y\in (-1,1)$, there holds $|f_n(x)-f_n(y)|\leq |x-y|^\alpha $ for some $\alpha\in (0,1)$ whenever $|x-y|>\frac{1}{3n}$. Then show that there exist $f\in C(-1,1)$ such that subsequence of $f_n$ converges unifomly to $f$ on $(-1,1)$ or give counterexapmle.
My attempt: I was trying to apply Arzela-Ascoli theorem. To show the uniformly equicontinuous I do not know how to control when $|x-y|<\frac{1}{3n}$. Also, I am not sure about the conterexample that such convergence can not happen.
Any help/hint in this regards would be highly appreciated. Thanks in advance!