Suppose $X_n$ and $Y_n$ are random variables defined on the same probability space. Prove that if $X_n$ converges weakly to a random variable $X$ and $Y_n$ converges weakly to a constant $c$, then $X_n Y_n$ converges to $cX$.
I'm new to weak convergence, and I've recently learned about theorems like Prokhorov's Theorem. I'm studying for an exam, and this is one problem that I have been stuck on for some time now. I saw a similar question on this website, but it was different since the convergence was not weak convergence.
I would greatly appreciate your help in solving this problem as I revise for my exams since I have been stuck for quite some time now.