Quite a small question. In Evan's PDEs he states the following theorem
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 exist functions $u_m\in C^\infty(\bar{U})$ such that $$u_m\to u \ \ \ \ in \ W^{k,p}(U)$$
I presume theis theorem will still hold if $u_m\in C^\infty_c(U)$. My main question is, does this require convergence for 'any' sequence? Or is it sufficient to have convergence for 'a' sequence.
The claim is no longer true in general if you replace $u_m \in C^\infty(\bar U)$ by the stronger condition $u_m \in C_c^\infty(\Omega)$. In fact, this is only possible for $u\in W^{k,p}_0(U)$, since this space is defined as the closure of $C_c^\infty(\Omega)$ in $W^{k,p}(U)$.
The intuition behind that is that functions in $W^{k,p}(U)$ and $C^\infty(\bar U)$ can have non-zero boundary values, which is not true for $C_c^\infty(\Omega)$.