I am studying myself some facts about $\alpha$-Hölder-continuous functions but I don't get any further by proving the following:
$(1)$ $\forall\alpha\in ]0,1]$ is $C^{0,\alpha}$ dense in $C^0(D)$ concerning the uniform norm and $D\subset\mathbb R^n$.
$(2)$ $\forall\alpha\in ]0,1]$ and compact set $K\subset\mathbb R^n$ is $(C^{0,\alpha}(K),||\cdot||_{C^{0,\alpha}(K)})$ a complete space (with $||u||_{C^{0,\alpha}(K)}:=||u||_{\sup}+\sup\limits_{{x,y\in K\space\&\space x\ne y}}\frac{|u(x)-u(y)|}{|x-y|^\alpha}$ and $C^{0,\alpha}(K)$ the space of all $\alpha$-Hölder-continuous functions)
$(3)$ All bounded closed subsets of $(C^{0,\alpha}(K),||\cdot||_{C^{0,\alpha}(K)})$ are compact.
So how do you prove one $(1),(2),(3)$ ?