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