Characterization of weak convergence

Question

Let $*$ be the convolution product and $\eta_n$, $\eta$ be random variables to $\mathbb{R}$ then we have: $$ \eta_n \mbox{ converges weakly to } \eta \Leftrightarrow \mbox{ for every contuously distributed } \varepsilon \mbox{ we have: }\\ (F_{\eta_n} * F_{\varepsilon})_n \mbox{ converges uniformly to } (F_{\eta} * F_{\varepsilon})$$ where $F_{\eta}$ is the cumulative distribution function of $\eta$. I would like to prove this statement or get a reference for a proof. What would be even more more awesome if this had a generalization for filters.