  A utility function \(U\) is bounded above if there is a limit to how good things can be according to \(U\), or more formally, if there is some least natural number \(sup\) such that for every \(A\) in \(U\)'s domain, \(U(A) \le sup\). Likewise, \(U\) is bounded below if there is a limit to how bad things can be according to \(U\), or more formally, if there is some greatest natural number \(inf\) such that for every \(A\) in \(U\)'s domain, \(U(A) \ge inf\). Expected utility theory can run into trouble when utility functions are unbounded above, below, or both.