Im currently proving that $(X_n)_n \subset L_1$ such $X_0 = 0$ and for every bounded stopping time $\tau$ $\mathbb E(X_\tau) = 0$ holds, then $(X_n)$ is a martingale (adapted to a discrete filtration $F_n$).
I have seen the proof but using the stopping time $\tau = n 1_A + (n+1) 1_{A^c}$ with $A \in F_n$. I used the time $\bar \tau = n 1_A $, is the previous a stopping time? I think yes, because $\{\bar \tau = m \}$ can be $A^c$ or $A$ (both in $F_n$) depending if $m = n$ or not.