Almost sure and Probability Convergence of the Maximum of I.I.D. Random Variables I am going over the Borel-Cantelli Problems in Durrett, but I am really stuck on Exercise 2.3.15.. 

Suppose $Y_{1},Y_{2},\ldots$ are i.i.d.. I need to prove: 
1) $\frac{\max _{1\leq m\leq n}Y_{m}}{n}\to 0$ in probability if and only if $nP(Y_{i}>n)\to 0$
2) $\frac{\max _{1\leq m\leq n}Y_{m}}{n}\to 0$ almost surely if and only if $E\left(\max\{0,Y_{i}\}\right)<\infty$.

There are two easier parts to the problem that I was able to solve. I proved that 
$$
\frac{Y_{n}}{n}\to 0\text{ in probability }\iff P(|Y_{i}|<\infty)=1
$$
and
$$
\frac{Y_{n}}{n}\to 0\text{ almost surely }\iff E|Y_{i}|<\infty.
$$
However, I am unsure if these facts can be used to solve the above two problems. I'd really appreciate some help with these problems.
Thanks!
 A: The statements you are trying to prove involve the maximum. For the first part, define $P_n:=\mathbb P\left(\max_{1\leqslant i\leqslant n}Y_i\gt n\varepsilon\right)$ for an arbitrary but fixed positive $\varepsilon$. Note that 
$$P_n=\mathbb P\left(\bigcup_{1\leqslant i\leqslant n}Y_i\gt n\varepsilon\right)=1-\mathbb P\left(\bigcap_{1\leqslant i\leqslant n}Y_i\leqslant n\varepsilon\right)$$
and since the events $\left(Y_i\leqslant n\varepsilon\right)_{1\leqslant i\leqslant  n}$ are independent and have the same probability, we derive that 
$$P_n=1-\left(\mathbb P\left(Y_1\leqslant n\varepsilon\right)\right)^n.$$
The equality $\mathbb P\left(Y_1\leqslant n\varepsilon\right)
=\left(1-P_n\right)^{1/n} $ entails $$n\mathbb P\left(Y_1\gt n\varepsilon\right)= n\left(1-\left(1-P_n\right)^{1/n}\right).$$
Now do an asymptotic expansion of $n(1-(1-t)^{1/n})$ to get that $P_n\to 0$ if and only if $n\mathbb P\left(Y_1\gt n\varepsilon\right)\to 0$. 
For the second part, one direction is done by the result you mentioned. For the converse, apply the Borel-Cantelli lemma to the sequence of events   $(A_n)$ defined by $A_n=\left\{\max_{1\leqslant i\leqslant n}Y_i\gt n\varepsilon \right\}$ (the probability is bounded by $n\mathbb P\left(Y_1\gt n\varepsilon\right))$. 
