"For a sequence of events $\{U_n\}_{n=1}^\infty$, let ${1}_{U_n}$ be and indicator for $U_n$. Let $N = \text{max}_{n \in \mathbb{N}}\{1_{U_n} = 1\}$. $\mathbb{P}(U_n \: i.o) = 0 \Longleftrightarrow \mathbb{E}[N] <\infty $."
I'm not sure if this is true in either direction. I tried forward first.
Suppose $\mathbb{P}(U_n \: i.o) = 0$. By Borel-Cantelli Lemma,
$\sum_{n=1}^\infty \mathbb{P}(U_n) < \infty $. Note that $\mathbb{P}(N = n) = \mathbb{P}(U_n)\mathbb{P}(\cap_{j=n+1}^\infty U_j^c) = \mathbb{P}(U_n)\mathbb{P}((\cup_{j=n+1}^\infty U_n)^c) = \mathbb{P}(U_n)-\mathbb{P}(U_n)\mathbb{P}(\cup_{j=n+1}^\infty U_n)$.
But then I don't know how to show that $\sum_{n=1}^\infty n\mathbb{P}(N=n) < \infty$ if it is a true statement.