Can anyone provide any insight in how I could show the following (attached below)
Hello,
$\displaystyle 1-F(x)=P(X>n)=P(X\cdot \bold{1}_{X>n}>n)=P(Y_n>n)\leq \frac{E(Y_n)}{n}$ by Markov's inequality.
where $\displaystyle Y_n=X\cdot \bold{1}_{X>n}$
As n goes to infinity, $\displaystyle Y_n$ obviously goes to 0 (X is almost surely finite since it's integrable)
and $\displaystyle |Y_n|$ is bounded by |X|, which is integrable.
So we can apply the dominated convergence theorem, and we have $\displaystyle n(1-F(x))\leq E(Y_n) \to 0 \quad \quad \square$