if X positive and E(X) < $\infty$ then find a limit

Question

if X a positive random variable and E(X)<$\infty$

how can we find/proof the result of the following limit $\lim_{x\to \infty}$ x P(X> x)

MY THOUGHT: $1)$ I was thinking that we can use something like this if $X>0$ we know $E(X)=\int_0^\infty dx \ P(X>x)$.

$2)$Also I was trying to find something with Dominated Convergence Theorem.

Answer 1

Hint:

1.Note $nP(X>n)\to 0$ as $n\to \infty,n\in\mathbb{N}$ iff $xP(X>x)\to 0$ as $x\to \infty,x\in\mathbb{R}$.

2.Let $f_n=n1_{\{X>n\}}$ then $f_n\to 0$ a.e. since $E(X)<\infty$ implies $X<\infty$ a.e.. Also $|f_n|\le X$. Use Dominated convergence theorem.