In Evans 275~277 trace-zero function theorem proof,
u $\in$ ${{W^1}^,}^p(R_+^n)$, $u$ has compact support in $\bar R_+^n$,
$Tu = 0$ on $\partial R_+^n$ = $R^{n-1}$
Then since $Tu = 0$ on ${{{R^n}^-}^1}$, there exist functions $u_m \in C^1(\bar R_+^n)$ such that
$u_m$ $\to$ u in ${{W^1}^,}^p(R_+^n)$ and $Tu_m=u_m|_{R^{n-1}} \to 0$ in $L^P(R^{n-1})$.
In here, my question is Why these $C^1(\bar R_+^n)$ functions exist