Linear continuous map

Usually I know that in order to show that a map is linear I have to prove the additivity and homogenity. In most cases, I worked with functions that were given. In the next exercise, it is a bit complicated the way it is formulated so I am not sure how to do things:

Let $u \in W_0^{1,p}(\Omega)$, $\Omega \in \mathbb{R}^n$, bounded, open and connected, $n<p< \infty$ and let $\alpha = 1 - \frac{N}{p}$ Take a sequence $u_n \ C_c^{\infty} (\Omega)$ with $u_n \rightarrow u$ in $W^{1,p}(\Omega)$. Prove that $u_n$ is a Cauchy sequence in $C^{\alpha}(\bar{\Omega})$, and that its limit $\bar{u}$ in $C^{\alpha}(\bar{\Omega})$ has the property that $|\bar{u}-u|_p = 0$. Prove that the map $u \rightarrow \bar{u}$ is linear and continuous from $W_0^{1,p}(\Omega)$ to $C^{\alpha}(\bar{\Omega})$.