Using Borel-Cantelli to find limsup 
Suppose $Y_1, Y_2, \dots$ is any sequence of iid real valued random variables
with $E(Y_1)=\infty$ . Show that, almost surely, $\limsup_n
(|Y_n|/n)=\infty$ and $\limsup_n (|Y_1+...+Y_n|/n)=\infty$.

I have solved the first part by considering non-negative integer iid r.vs $X_n=\text{floor}(|Y_n|)$ and using $E(X)=\sum_0^\infty P(X\ge n)$ then doing some clever tricks so I can apply the (2nd) Borel-Cantelli lemma, but I'm not really sure how I can use the same approach to solve the second part seeing as it is tempting to set $S_n=\text{floor}(|Y_1+...+Y_n|)$ but then the $S_i$ are not iid. I'm pretty sure its gonna be  Borel-Cantelli again (since limsup) so I need to come up with the right events. Please can someone nudge me in the right direction.
Hints only please
EDIT:
Suppose $\limsup_n |a_1+...+a_n|/n\lt \infty$. Then set $S_n=\sum^n_1 a_k$ $$\frac{|a_n|}{n}=\frac{|S_n - S_{n-1}|}{n}\\ \le\frac{|S_n|}{n}+\frac{|S_{n-1}|}{n-1} $$ bounded

Hint: If $(a_n)$ is a sequence of real numbers with $\lim \sup {|a_1+a_2+\cdots+a_n|}/n <\infty$ then $\lim \sup \frac {|a_n|} n <\infty$.
