I'm trying to show that $D(\Omega)$, i.e. the classical space of test functions, is not dense in $H^1(\Omega)$. To show this, I consider the closure of $D(\Omega)$ in the $H^1$ norm, call it $D$.
Let $u \in D$. By definition of closure, there exists a sequence $\{u_k\}_k \in D(\Omega)$ such that $||u_k -u||_{H^1} \rightarrow 0$. If now I consider the classical trace operator $\gamma : H^1(\Omega) \rightarrow L^2(\partial \Omega)$, by its continuity I have: $$\gamma(u_k) \rightarrow \gamma(u)$$
But since $\gamma(u_k)=0$, it follows $\gamma(u)=0$. Therefore, in $D$ every function has $0$ trace, and then a generic function in $H^1$ with non-zero trace cannot be reached as a limit in $H^1$ of elements of $D$. Is that correct?