I was inspired by this question and Meyers-Serrin theorem to prove that there exists a function $u \in W^{1,\infty}(I) $ such that there are no functions $u_m \in C^{\infty}(I) \cap W^{1,\infty}(I)$ with $u_m \to u$ in $W^{1,\infty}(I)$.
Generally speaking, I'm trying to prove that $C^{\infty}(I) \cap W^{1,\infty}(I)$ is not dense in $W^{1,\infty}(I)$. This leads me into more difficulties than I supposed. Hints?
Moreover, what is the closure of $C^{\infty}(I) \cap W^{1,\infty}(I)$ wrt $W^{1,\infty}(I)$-norm?
Here $I$ is only an open interval of the real line. I would try to generalize this fact by myself.