Products and exponentials of an a.s.-vanishing sequence stay degenerate at their limits

Let $\{X_n,n\ge1\}$ be a sequence of independent identically distributed random variables with mean 0 and finite variance $\sigma^2>0$. Let $S_n=\dfrac{X_1+X_2+\dots+X_{n^2}}{n^2}$. Then which of the following is false?