Let $X\_1,...,X\_n$ are independent identical random variables of standard Cauchy distribution (i.e. the density is $\frac{1}{\pi(1+x^2)}$). Let $n \to \infty$ and consider the random variable $S\_n=X\_1+,...,+X\_n$.

1. Prove: $\frac{S\_n}{a\_n}\rightarrow0$ a.e. iff $\sum \frac{1}{a\_n}<\infty$.
2. Try to give the necessary and sufficient conditions for which $$
\frac{\max(X\_1,...,X\_n)}{a\_n}\to 0 \quad\text{a.e.}
$$

**My ideas so far**:

1. For problem 1, given $\sum \frac{1}{a\_n}<\infty$, I would like to prove that $\frac{S\_n}{a\_n}\rightarrow0$: by Kroencker lemma, I just need to prove that $\sum\frac{X\_n}{a\_n}<\infty.$ Then I do it by applying Kolmogorov three series theorem: thus I can easily prove the convergence of $\sum\frac{X\_n}{a\_n}$. But how to prove the converse? What can I deduce from $\frac{S\_n}{a\_n}\rightarrow0$? Maybe I should try to use characteristic function?
2. For problem 2, I think maybe there is some way to convert $\frac{\max(X\_1,...,X\_n)}{a\_n}$ to a similar form as in problem 1: am I right?

Thanks in advance for any tips or help in general.