Let $X_n$ be independent random variables such that $X_1 = 1$, and for $n \geq 2$,
$P(X_n=n)=n^{-2}$ and $P(X_n=1)=P(X_n=0)=\frac{1}{2}(1-n^{-2})$.
Show $(1/\sqrt{n})(\sum_{m=1}^{n}X_m-n/2)$ converges weakly to a normal distribution as $n \rightarrow \infty$.
I believe I will be using Central Limit Theorem for this problem, but I am not sure how to apply it. Help/hints would be appreciated!