Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 38 / 6 / d

Part (c) gives $\mathbb{E}{M}_t\le 0$. Since $Y_t{X}_t\ge 0$, its integrated identity impliesfor every finite $t$. The cumulative consumption increases with $t$, so the monotone convergence theorem yieldsThis is the infinite-horizon state-price budget constraint. No terminal-wealth convergence or vanishing assumption is needed for this inequality.