# Lim sup and lim inf of IID RVs

$$X_1, X_2, \ldots$$ are iid random variables with $$P ( X_1 = n ) = P ( X_1 = - n ) = \frac{ c }{ n^2 \log n }$$ where c makes the probabilities sum to one. Define $$S_n = X_1 + \ldots + X_n$$. We want to show that
$$\limsup \frac{S_n}{n} =\infty$$ and $$\liminf \frac{S_n}{n} = -\infty$$ almost surely.

I've managed to use the Borel-Cantelli lemma to show that $$P(|X_n| \geq n \text{ infinitely often}) = 1$$, but I can't pass to the lim sup/inf. Any help/suggestions would be appreciated.