I would like to ask for help with proving the finiteness of the following double integral.
$$\int_{0}^{\infty}e^{\alpha+k}\int_{k+\zeta}^{\infty} (e^{-\zeta+x}-e^k)f(x)\ \mbox{d}x\ \mbox{d}k,$$
where $\alpha>0$, $\zeta\in\mathbb{R}$ and $f$ is probability density function such that $\int_{-\infty}^{\infty}e^{-\zeta+x}f(x) \mbox{d}x<\infty$.
Thanks in advance.