Let $X_1, X_2, ...$ be a sequence of random variables such that:
$X_i = \begin{cases} 1 &\mbox{with probability } p = 1-1/n^2 \\ 2 & \mbox{with probability } p = 1/n^2 \end{cases} $
and let $Y_n=\Pi_{i=1}^n X_i $. Show that $Y_n$ converges almost surely to some random variable $Y$.
I already know several methods of proving a.s. convergence to a constant. However, I do not know how to establish convergence to a non-constant random variable except for definition which does not seem to work in this case.