Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 4 / b / Solution

For each integer $m$, stop on first crossing below $-m$. Bounded increments ensure the stopped martingale is bounded below by $-m-C$; after adding $m+C$ it is a nonnegative supermartingale and therefore converges. Thus on the event that $(X_n)$ is bounded below, it converges finitely. Applying the same argument to $-X_n$ shows that boundedness above also forces convergence. Outside the finite-limit event the path is therefore unbounded in both directions, so its limsup is $+\infty$ and its liminf is $-\infty$. Hence $\mathbb{P}(A\cup B)=1$.