Let $\Omega\subset\mathbb R^n$ be compact and $C^{0,\alpha}(\Omega)$ the space of all $\alpha$-Hölder-continuous functions. Define $||u||_{C^{0,\alpha}(\Omega)}:=||u||_{\sup}+\sup\limits_{{x,y\in \Omega\space\&\space x\ne y}}\frac{|u(x)-u(y)|}{|x-y|^\alpha}$ and consider $(C^{0,\alpha}(\Omega),||u||_{C^{0,\alpha}(\Omega)})$ and $\alpha\in]0,1]$ .
How can you prove that for any sequence in bounded closed set of $(C^{0,\alpha}(\Omega),||u||_{C^{0,\alpha}(\Omega)})$ there exists a convergent subsequence (concerning the uniform norm) and it limes is in $(C^{0,\alpha}(\Omega))$?