Let $B\subset \mathbb{R}^n$ be a open ball. Let $\{f_i\}$ be a sequence of functions bounded in the norm $C^{k,\alpha}(B)$ for  given integer $k\geq 0$ and $\alpha\in (0,1)$.

>Does there exist a subsequence which converges to a function $f$ (necessarily $f\in C^{k,\alpha}(B)$) in the norm $C^{k,\alpha/2}(B')$ for any smaller subball $B'\subset B$? 

A reference would be very helpful.