Suppose U : \mathbb{R} \to \mathbb{R} is concave, and that the random variable \epsilon has zero mean. Assuming that the function \phi : \mathbb{R} \to \mathbb{R} , defined by \phi(\lambda) = \mathbb{E} U(\mu + \lambda \epsilon) is everywhere finite-valued, prove that \phi is concave.

I've tried a few different things, including Jensen's inequality, but I can't get it to work. Any help would be greatly appreciated. Thanks