Utility function $u(x)$ is monotonic. I want to prove that $u(x)$ exhibits risk aversion if and only if for all lottery $F$: $E(x) \geq CE(F,u)$ (CE is certainty equivalent).
(Definition of $CE$: the certainty equivalent $CE(F,u)$ of lottery $F$ given Bernoulli utility $u(x)$ is such that $u(CE(F,u)) = E(u(x)) = U(F)$)
I tried with
$u(x)$ is risk averse $<=>$ $u(x)$ concave $<=>$ $E(u(x)) \leq u(E(x))$ (Jensen's inequality)
I am not sure about how to proceed. Since $u(x)$ is monotonic and not strictly monotonic, I can not take inverse of $u(x)$ on both side of the inequality.