Rare large jumps break Lindeberg: the normalised sum is not Gaussian

Let $\{X_n\}$ be a sequence of (independent) random variables such that $P(X_n=-n^{1/2})=P(X_n=n^{1/2})=1/2n$ and $P(X_n=0)=(n-1)/n$. Let $S_n=\sum_{i=1}^nX_i$. Then, which of the following statements is not true?