This statement was just thrown out there in a proof for Cramers Large Deviation Theorem:
If $\varphi_X(\theta) = E(e^{\theta X}) < \infty$ on an open neighborhood of $\theta_0$ then $$E(\lvert X\rvert^p e^{\theta X})<\infty,\quad 0\le p \in\mathbb{R}$$ I don't see why this is obvious, and when I tried to look at the integral for the expected value, it didn't get any clearer.
I don't really have any ideas, so little is known about $X$ (it's just a real-valued RV).
How can I show this result?