Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 39 / 5 / b / Solution

A unit zero-coupon bond pays one at its maturity, so $P_t(t)=1$. Monotonicity in maturity gives $P_t(t+1)\le 1$. Using the state-price density representation with $T=t+1$,The process is positive and integrable, as noted in part (a), and adapted. Thus it is a supermartingale. Strictly decreasing maturity prices yield a strict one-step conditional inequality; weak decrease is already enough for the conclusion.