= Solution
Let $(M_t)_{t\geq0}$ be a right-continuous <continuous-time martingale> with $\sup_t\mathbb E|M_t|<\infty$. Restricting it to the nonnegative rational times gives a countable martingale. The argument of part (b), applied on successively finer rational grids, shows that it has a finite almost-sure limit as the rational time tends to infinity. <Right-continuous function>[Right-continuity] and the upcrossing characterization prevent the values at arbitrary times from having a different limit. Thus $M_t$ converges almost surely to a finite integrable random variable as $t\to\infty$, which is the <Continuous-time martingale convergence theorem>.
Back to article page
