Let $S_{n}:= \sum_{i=1}^{n} X_{i,n}$ where for each $n, X_{1,n}, X_{2,n},..., X_{n,n}$ are sequences of independents r.v.'s. $$X_{i,n}=\begin{cases}1, & \text{with probability }p_n\\0,& \text{with probability }1-p_n\end{cases}$$ for $i = 1,\dots, n$. Suppose that $p_{n}={\beta}/n$.
The problem is to show that $S_{n}$ doesn't converge almost surely to a limit.
I proved that it converges in distribution to the Poisson, but I'm not sure how to proceed in proving that $S_{n}$ does not converge almost surely to a limit. The hint given is to show that $ P(|S_{m}-S_{n}|>1) $ does not converge to zero. I know this is a Cauchy so possible giving me guidance in this directions. (or any other that you see fit)??
Thanks!!