If you have a sequence of random variables $(W_n)$ which converges in distribution to N, and another sequence of random variables $(X_n)$ which converges in distribution to $B$:
i) Will $(W_nX_n)$ converge in distribution to N multiplied by B?
ii) Similarly, will $(W_n + X_n)$ converge in distribution to $N + B$?
I know this is probably a very basic question, but I can't find proofs/discussions properties of convergence in my text books :(
If it's not too complicated, could you let me know whether i and ii happen to be true (or false) for the other basic types of convergence (almost-sure, L^2, pointwise) ?