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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 $C^{0,\gamma}(B_1)$.

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Set $u_k^+(x) = k + \frac{1}{2k}$ and $u_k^-(x) = k - \frac{1}{2k}$. Then $\Vert u_k^+ - u_k^-\Vert = 1/k$, but no limit exists.

You need boundedness of one of the sequences in $C^{0,\gamma}$.