I suspect the following result is true but I"m not sure how to go about proving:
It is given that $\Omega \subset \mathbb{R}^{n}$ is an open bounded, connected domain.(Not sure if theses conditions on the domain are important)
If $\{v_{n}\}_{n} \subset C_{c}^{\infty}(\Omega)$ such that $v_{n} \rightarrow v$ in $L^{1}(\Omega)$. Does it follow that for $f \in C(\Omega)$($f$ is continuous) that $fv_{n} \rightarrow fv$ in $L^{1}(\Omega)$?
thanks for any assistance