Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 4 / a / Solution

Localize when the predictable integrand first exceeds level $n$, just before that increment is taken. The resulting stopped integrand is bounded, so its martingale transform is a martingale by the stated result. The localization times increase to infinity because each finite collection of $H_t$ is finite almost surely. Hence $X$ is a local martingale.