Exercise from Williams book Probability with martingales

Question

I'm doing this exercise from Williams book Probability with martingales

Let $(X_n)$ be a sequence of IID random variables with $E(|X_n|) = \infty $ for all $n$. Then prove that $$1)\ \sum_n P(|X_n| > kn) = \infty \ \ k \in \mathbb{N}$$ $$2)\ \limsup_n \frac{|X_n|}{n} = \infty \ \ a.e. $$ $$3) \ \limsup_n \frac{|S_n|}{n} = \infty \ \ a.e. $$ where $S_n = X_1+ \ldots + X_n$.

I'm stuck with the point $3$, how to proceed ?

Answer 1

Suppose that

$$\limsup_{n \to \infty} \frac{|S_n(\omega)|}{n}<\infty$$

for some $\omega \in \Omega$. Then

$$X_n = S_n-S_{n-1}$$

implies

$$\begin{align*} \limsup_{n \to \infty} \frac{|X_n(\omega)|}{n} &\leq \limsup_{n \to \infty} \frac{|S_n(\omega)|}{n} + \limsup_{n \to \infty} \frac{|S_{n-1}(\omega)|}{n} \\ &= \limsup_{n \to \infty} \frac{|S_n(\omega)|}{n} + \limsup_{n \to \infty} \frac{n-1}{n} \frac{|S_{n-1}(\omega)|}{n-1} < \infty. \end{align*}$$

Now the claim follows from 2).