Expected utility representation

Question

I am stuck on some question on utility theory. The question is as follow:

Consider $A=[0,+\infty)$, and $Q=${F-cumulative distribution function on $A: \int^{+\infty}_0 x dF(x)<\infty$}, the set of distributions of random variables with finite expectation. I am asked to show if an agent's preference is represented by expected utility which is consistent with the axioms of rational choice, then the utility function u must satisfy

$\lim_{x\rightarrow{\infty}} u(x)/x\leq c $

for some constant c.

There is a hint saying that suppose the condition on u(x) is not satisfied, then show there exists a distribution in $Q$ which is inconsistent with the Archimedean axiom.

Thanks so much.

Answer 1

1) Show that if the condition does not hold for $\hat u$ there must exist some $F\in Q$ such that $$\hat U(F) =\int_0^\infty \hat u(x)dF(x)=+\infty.$$

2) Prove that $\hat U(F)=+\infty$ violates the Archimedean axiom.