Suppose that $X$ is an essentially bounded variable, $\mathbb{E}X=0$, prove that $\mathbb{E}e^X\leq\cosh \|X\|_\infty$.

Question

Problem:

Suppose that $X$ is an essentially bounded random variable, $\mathbb{E}X=0$, prove that $\mathbb{E}e^X\leq\cosh \|X\|_\infty$.

Here $X$ isn't necessarily a discrete or continuous r.v., but if we prove it for discrete and continuous r.v., can we get that it holds for all r.v.?

I tried to prove it using integral by parts, supposing that $X$ is continuous, but nothing was found...