Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 202 / 5 / ii / Solution

Take the stopping timesContinuity gives $|M_{t\wedge S_n}|\le n$, with no overshoot at the first crossing. Stopping preserves the local martingale property, and the bounded case proved in question 4(i) makes each $M^{S_n}$ a continuous true martingale.

Every continuous path is bounded on each compact time interval, so $S_n\uparrow \infty$ almost surely. Hence this is also a localizing sequence consisting of bounded stopped martingales.