The following is an exercise in Prof Tao's lecture notes on probability theory.
We assume that a sequence of random variables $\xi_n \rightarrow \xi$ in distribution and also $\nu_n \rightarrow \nu$ in distribution as well.
(i) If $\nu$ equals a constant almost surely, then prove that the product $\xi_n \nu_n$ converges to $\xi \nu$ in distribution.
(ii) Find a counterexample to (i) in the case $\nu$ does not equal a constant a.s.
Any help, hints will be greatly appreciated.