First one:

Suppose $\displaystyle X_n \sim Bin(n, p)$. Show that as $\displaystyle n \rightarrow \infty$

$\displaystyle

\frac{X^2_n - n^2 p^2}{2(pn)^\frac{3}{2} \sqrt{1-p})} \stackrel{D}{\rightarrow} Z \sim N(0,1)

$

In a previous question I proved that $\displaystyle \frac{X_n - np}{\sqrt{np(1-p)}} \stackrel{D}{\rightarrow} Z \sim N(0,1)$, so for this one, do I just have to show that the left hand side is $\displaystyle X^2_n$ 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 $\displaystyle X^2_n$ so I'm not sure if that's the way to go.

Second one:

Show that $\displaystyle (1-X_n)^{-1} \stackrel{D}{\rightarrow} (1-X)^{-1}$ given that $\displaystyle X_n \sim P(X_n = \frac{i}{n}) = \frac{1}{n}, i = 0, 1, 2, 3, .. n-1$

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

$\displaystyle

E(e^{\lambda X_n}) = \sum_{x=\frac{i}{n}}^{\frac{n-1}{n}} e^{\lambda x} P(X=x)

$

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?