For continuous random variables:
$E(U'(\xi)) = \int_0^{+\infty} U'(x) f_\xi(x)dx < +\infty$, because $U'(+\infty)=0$, and boundness of $U'$ is suffucuent here.

Then, $\xi U'(\xi) - U'(\xi) \leq U(\xi) - U(1)$, because $U$ is concave, and $E(\xi U'(\xi)) \leq E(U(\xi)) + E(U'(\xi)) - U(1) < +\infty$.