Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 4 / a / Solution

Continuity gives $X_{{\tau }_a}=a$ on $\{{\tau }_a<\infty \}$. The stopped process $X^{{\tau }_a}$ is bounded by $a$ and is therefore a true martingale. HenceThe second term tends to zero by bounded convergence because $X_t\to 0$ and it is bounded by $a$. Thus