Suppose that $u_{k}^{+}, u_{k}^{-} \in C^{0,\gamma}(B_1)$ where $B_1$ is the unity ball in $\mathbb{R}^{n}$ (for some $\gamma \in (0,1)$, the Holder space) is a sequence such that $\|u^{+}_{k} - u_{k}^{-}\|_{C^{0,\gamma}(B_1)} \le 1/k$, Can we assume, up to subsequence that $u_{k}^{+}, u_{k}^{-}$ converge to a $u$?
I can see that if $u_{k}^{+}$ is a sequence bounded in $C^{0,\gamma}(B_1)$ we can use Ascoli-Arzelá and obtain this. But if
I can see that if $u_{k}^{+}$ is a sequence bounded in $C^{0,\gamma}(B_1)$ we can use Ascoli-Arzelá and obtain this. But $\|u^{+}_{k} - u_{k}^{-}\|_{C^{0,\gamma}(B_1)} \le 1/k$ seens not implies that $u_{k}^{+}$ is bounded in $C^{0,\gamma}(B_1)$.