From my reply to komponisto (incidentally, both you and he seem to be making the same objections in parallel, which suggests that I’m not doing a very good job of explaining myself, sorry):
Suppose that there is nothing so bad that you would pay $5 to stop it from happening with probability 10^(-100). Let X be a state of the universe. Then u(-$5) < 10^(-100) u(X), so u(X) > 10^(100) u(-$5). Since u(X) > 10^(100) u(-$5) for all X, u is bounded below.
Similarly, suppose that there is nothing so good that you would pay $5 to have a 10^(-100) chance of it happening. Then u($5) > 10^(100) u(X) for all X, so u(X) < 10^(100) u($5), hence u is also bounded above.
From my reply to komponisto (incidentally, both you and he seem to be making the same objections in parallel, which suggests that I’m not doing a very good job of explaining myself, sorry):