TITLE: Does $\mathbb{E}\left[X\right]=\infty$ imply $\mathbb{E}\left[X^{2}\right]=\infty$?
QUESTION [8 upvotes]: I'm trying to prove that if $\mathbb{E}\left[X\right]=\infty$ then $\mathbb{E}\left[X^{2}\right]=\infty$ for every random variable $X$.
I know that if $X(w)>1$ I'll get that $X^2(w)>X(w)$ so $\mathbb{E}\left[X^{2}\right]\ge\mathbb{E}\left[X\right]$
and if $X(w)\le1$ then $X^2(w)\le X(w)$  so
$\mathbb{E}\left[X^{2}\right]=\sum\nolimits _{w\in\Omega}X^{2}\left(w\right)\cdot p(w)\le\sum\nolimits _{w\in\Omega}1\cdot p(w)=1$
But how do I prove it when some of the $w\in \Omega$ are larger than 1 and some are less ?
Is this even the right way to prove this ?

REPLY [2 votes]: Note that
$$
0\leq\text{Var} X=E(X-EX)^{2}=EX^2-(EX)^2
$$
so that
$$
(EX)^2\leq EX^2.
$$
Hence $$EX=\infty \implies EX^2=\infty.$$