In Evans' book, theorem 5.3.3
THEOREM 3 (Global approximation by functions smooth up to the boundary).
Assume $U$ is bounded and $\partial U$ is $C^1$. Suppose $u\in W^{k,p}(U)$ for some $1\leq p<\infty$. Then there exxists functions $u_m\in C^\infty(\bar{U})$ such that $$u_m\to u \quad\text{in }W^{k,p}(U)$$
For some $x^0\in \partial U$, by $C^1$ boundary, there exists $r>0$ such that $U\cap B(x^0,r)=\{x\in B(x^0,r) : x_n>\gamma(x_1,\dots,x_{n-1})\}$.
In the proof, Evans defined $V=U\cap B(x^0,\frac{r}{2})$, and the upshifted version of $u_\epsilon(x):=u(x^\epsilon)$ where $x^\epsilon:=x+\lambda\epsilon e_n$, where $\lambda>0$ is chosen such that for small enough $\epsilon>0$, $B(x^\epsilon,\epsilon)\subset\subset U\cap B(x^0,\frac{r}{2})$.
From before, Evans shows that for any $u\in W^{k,p}(U)$, $$||\eta_\epsilon \ast D^\alpha u-D^\alpha u||_{L^p_{\text{loc}}(U)}\to 0$$
My convern arise at step 3 of his proof, he claimed for all $|\alpha|\leq k$, the term
$$||\eta_\epsilon \ast D^\alpha u_\epsilon(x)-D^\alpha u_\epsilon(x)||_{L^p(V)}\to 0 \quad\text{as }\epsilon\to0$$
This can be rewritten as
$$||D^\alpha u(x)-D^\alpha u(x)||_{L^p(V+\lambda\epsilon e_n)}\to 0 \quad\text{as }\epsilon\to0$$
The problem here is that I don't know if the previously proven claim can be used on this moving region. I have tried to find a compact subset of $U$ that contains $V+\lambda\epsilon e_n$ for all $\epsilon>0$, but that leads to no where. I have shown that $V+\lambda\epsilon e_n\subset U_\epsilon:=\{x\in U: dist(x,\partial U)>\epsilon\}$, but surely no compact subset of $U$ contains $U_\epsilon$ for all small enough $\epsilon>0$.
How do I resolve this?