Solution
= Solution
Part (c) gives $\mathbb E M_t\leq0$. Since $Y_tX_t\geq0$, its integrated identity implies
$$
\mathbb E\int_0^tY_sc_sds\leq Y_0X_0
$$
for every finite $t$. The cumulative <consumption> increases with $t$, so the <monotone convergence theorem> yields
$$
\boxed{\mathbb E\int_0^\infty Y_tc_tdt\leq Y_0X_0.}
$$
This is the infinite-horizon <state-price budget constraint>. No terminal-wealth convergence or vanishing assumption is needed for this inequality.