Embedding of differentiable functions into Holder Functions

Question

I have the following problem. Consider $\Omega \subset \mathbb{R}^{n}$, fix $\alpha \in (0,1) $. Consider the space $C^1(\overline \Omega)$ of differentiable functions equipped with the norm $${\| u \|}_{1} = \sup_{x\in \overline \Omega} |u(x)| + \sum_{k=1}^{n} \sup_{x\in \overline \Omega} \left\lVert \frac{\partial u(x)}{\partial x_k} \right\rVert $$ I need to show it is compactly embedded in the space $C^{\alpha}(\overline \Omega)$, with its norm being $${\| u \|}_{\alpha} = \sup_{x\in \overline \Omega} |u(x)| + \sup_{x,y \in \overline\Omega,\, x \neq y} \frac{\lvert u(x)-u(y) \lvert}{\rvert x-y \rvert ^{\alpha}} $$
I know I have to prove that there exists a constant $c >0$ such that ${\| u \|}_{\alpha} \le {c\| u \|}_{1} $ and every bounded set in $C^{1}$ is precompact in $C^{\alpha}$. However, in the first place I find it difficult to get such a constant. Then, I was trying to use the Ascoli-Arzelà theorem: is it a good strategy? how do I get the uniform-continuity and uniform-boundedness?
P.S. Is it necessary to consider $\Omega$ bounded or may I have the same results for any open $\Omega \subset \mathbb{R}^{n}$?