Let $X_1,X_2,X_3,...$ be i.i.d. with $P(X_1 >0)=1$. Define $S_n =\Sigma_{i=1}^{n} X_i$.
(a) Suppose $\mathbb{E}[X_1] <\infty$. Show that $\lim_{n\rightarrow \infty} \frac{X_n}{S_n}=0$ a.s.
I think here $\frac{X_n}{S_n}=\frac{n}{S_n}\times \frac{X_n}{n}$ and $\frac{X_n}{n}\rightarrow 0$ because here $X_n$ is a $L^{1}$ function. I don't think it's rigorous, or may be even wrong.
(b) Constrct an example in which $\mathbb{E}[X_1] =\infty$ and $\lim_{n\rightarrow \infty} \frac{X_n}{S_n}=0$ a.s is false.
No idea about the second part.