Local martingales in discrete time

For any discrete-time $P$--local martingale $S$ there exists a probability measure $Q \sim P$ such that $S$ is a $Q$--martingale. A new proof for this result is provided. The core idea relies on an appropriate modification of an argument by Chris Rogers, used to prove a version of the fundamental theorem of asset pricing in discrete time. This proof also yields that, for any $\varepsilon>0$, the measure $Q$ can be chosen so that $\frac{dQ}{dP} \leq 1+\varepsilon$.