Let $X$ be a discrete local martingale such that $X_T$ is integrable for some non-random time $T > 0$. I am tasked with showing that $(X_t)_{0≤t≤T}$ is a true martingale.
The hint is to show that $X_{T-1}$ is integrable, and I think that the result would follow by induction. However, I really can't think where to start with achieving this. Any help would be greatly appreciated, thanks!