I'm having trouble with an exercise (E4.6 Converse to SLLN) in "Probability with Martingales" by David Williams. The problem is as follows:
Let $Z$ be a non-negative RV. Let $Y$ be the integer part of $Z$. Show that $$Y=\sum_{n\in \mathbb{N}}I_{\{Z\ge n\}},$$ and deduce that $$\sum_{n\in \mathbb{N}}P[Z\ge n]\le E[Z]\le 1+\sum_{n\in \mathbb{N}}P[Z\ge n].$$ Let $(X_n)$ be a sequence of IID RVs (independent, identically distributed random variables) with $E[|X_n|]=\infty$, $\forall n$. Prove that $$\sum_n P[|X_n|>kn]=\infty\ \ (k\in \mathbb{N})\ \ \text{and}\ \ \lim \sup \frac{|X_n|}{n}=\infty,\ \ \text{a.s.}$$ Deduce that if $S_n=X_1+X_2+\cdots+X_n,$ then $$\lim \sup \frac{|S_n|}{n}=\infty,\ \ \text{a.s.}\phantom{}$$
I have trouble with the last part about $S_n$. I was trying to relate $P[|S_n|>kn]$ with $P[|X_n|>kn]$ but I don't know how. I might not be on the right track. Could anybody take a look at this? Thanks!
From $${X_n\over n}={S_n\over n}-\left({n-1\over n}\right){S_{n-1}\over n-1},$$ we get $${|X_n|\over n}\leq {|S_n|\over n}+\left({n-1\over n}\right){|S_{n-1}|\over n-1},$$ and so $$\limsup_n {|X_n|\over n}\leq 2\limsup_n {|S_n|\over n}.$$ Therefore $\limsup_n |X_n|/n=\infty$ implies $\limsup_n |S_n|/n=\infty$.