I want to prove that for $u \in W_0^{l,p}(\Omega)$ we have convergence of mollifiers $u_\rho \rightarrow u$.
I appreciate that $u_\rho \rightarrow u$ in $L^p(\Omega)$ so just need to show convergence of the derivatives, i.e.,
$$ D^\alpha u_\rho \rightarrow D^\alpha u $$ in $L^p(\Omega)$.
My thoughts are to use a sequence $u_m \rightarrow u$ in $W_0^{l,p}(\Omega)$ with $u_m \in C_0^\infty (\Omega)$ with some form of triangle inequality, though I'm not exactly sure how. I know that $D^\alpha u_\rho = (D^\alpha u)_\rho$ in compact subsets of $\Omega$. This may help with $u_m$ compactly supported in $\Omega$.
Thanks.