I'm working on Durrett, exercise 2.3.15.
Let $(Y_n)_{n \geq 1}$ be sequence of iid rvs. I want to show that
$Y_n /n \rightarrow 0$ a.s. $\Leftrightarrow$ $E|Y_1| < \infty$
What I have tried: $$Y_n /n \rightarrow 0 \text{ a.s.} \Leftrightarrow \\ \forall \varepsilon \gt 0, P(|Y_n| \geq n\varepsilon \ \text {i.o})=0 \Leftrightarrow\\ \forall \varepsilon \gt 0, P(|Y_n| \lt n\varepsilon \ \text {ev})=1 \Leftrightarrow\\ \forall \varepsilon \gt 0, lim_{n \to \infty}P(\bigcap_{k \geq n}|Y_k| \lt k\varepsilon )=1 \ \Leftrightarrow\\ \forall \varepsilon \gt 0, lim_{n \to \infty}\prod_{k \geq n}P(|Y_k| \lt k\varepsilon )=1 \Leftrightarrow\\ \forall \varepsilon \gt 0, lim_{n \to \infty}\prod_{k \geq n}P(|Y_1| \lt k\varepsilon )=1$$
which is a dead end. Any help is appreciated.
Hint For any non-negative RV X, we have $$E(X)\le \sum_{n=1}^\infty P(X\ge n) < E(X) +1.$$ This, combined with Borel-Cantelli, will give you your desired equivalence between $P(Y_n > n\epsilon\;i.o.)=0$ and $E(|Y|) < \infty.$