Lim sup and lim inf of IID RVs

[rochfor1]

$$X_1, X_2, \ldots$$ 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.

 

[Enuma_Elish]

Suppose limsup S(n)/n is not infinity; instead limsup S(n)/n = m where m is finite...

 

[gel]

Note that if S(n)/n were bounded then X(n)/n = S(n) - (1-1/n)S(n-1) would also be bounded. Can you show that this is false?
Once that's done, Kolmogorov's 0-1 law should finish it off.