 To explain why this axiom entails that no object can have infinite value, suppose for reductio that A is a prize check worth $1, B is a check worth $2, and C is a prize to which the agent assigns infinite utility. The decision maker’s preference is \(A\prec B\prec C\), but there is no probability p such that \(\{pA, (1-p)C\sim B\). Whenever p is nonzero the decision maker will strictly prefer \(\{pA, (1-p)C\}\) to B, and if p is 0 the decision maker will strictly prefer B. So because no object (lottery or outcome) can have infinite value, and a utility function is defined by the utilities it assigns to those objects (lotteries or outcomes), the utility function has to be bounded.