A problem on almost sure convergence of an average I have the following exercise:

Let $X_1, X_2 \ldots$ be such that   $$ X_n = \left\{
  \begin{array}{ll}   n^2-1  & \mbox{with probability } n^{-2} \\   -1
 & \mbox{with probability } 1-n^{-2}  \end{array} \right. $$ Show that
  $S_n/n \rightarrow -1$ almost surely.

(Here $S_n = \sum^n X_i$)
So I see that the set in which $X_n=-1$ eventually will have measure $1$, i.e.
$$
P(\lbrace \omega: X_n(\omega)=-1 \rbrace) \rightarrow 1
$$
Now one has to show that 
$$
P(\lbrace \omega: S_n/n=-1 \rbrace) \rightarrow 1
$$
I have tried doing stuff like defining $Z_n = (n^2-1)1_{ \lbrace X_n=n^2-1 \rbrace}-1_{ \lbrace X_n=-1 \rbrace}$ and using Markov inequality but my calculations lead to $S_n/n \rightarrow 0$.
Any hints are much appreciated. 

Hint: Apply the Borel-Cantelli lemma to the sequence of events
$$A_n := \{\omega \in \Omega; X_n(\omega) \neq -1\}, \qquad n \in \mathbb{N}.$$
