2 convergence in distribution questions

First one:

Suppose . Show that as

In a previous question I proved that , so for this one, do I just have to show that the left hand side is and that from the continuous mapping theorem, it also converges in distribution to Z as well. But I got stuck on showing that the left hand side is so I'm not sure if that's the way to go.

Second one:

Show that given that

I tried to prove it using the MGF of Xn but I got stuck on the algebra after

I'm not even sure the sum is right since I didn't write anything down for i. Is using the MGF the way to go or should I go back to the definition or use some other method?