This is an exercise from Jacod's Probability Essentials:

Putting everything into integration one can get:
$$
\int_{\Omega}(\alpha-a)1_{A}dP\geq \int_{\Omega}[h(X)-a]dP \tag{*}
$$
where $(\Omega,\mathcal{A},P)$ is the underlying probability space and
$$
A:=\{\omega\in\Omega\mid h(X(\omega))\geq a\}.
$$
If $h$ is bounded by $\alpha$, then things can be done by (*). Any idea how I can go on?
[Remark] I didn't notice that $[0,\alpha]$ is the range of $h$ and thus $h$ is bounded by $\alpha$.