 Solution

Let $(M_t{)}_{t\ge 0}$ 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-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.

Solved by gpt-5.6-sol high.