When the global optimal is outside of the constraint set, what will be the demand?
Argue that, given your assumptions on the utility function, $x^*$ is the essentially unique (and hence global) maximum. (You need this because there may be local maxima when the assumptions on the utility function is relaxed - this will violate the proposition you're trying to prove).

Now simply use the definition of global optima: for any $x\leq x^*$, $u(x)\leq u(x^*)$. This should be enough to give you the result.