I want to prove the optional sampling theorem using the fact that discrete stochastic integrals for martingale integrators are still martingale.
To prove: if $(M_t)_{t\in T}$ is a martingale and $\sigma,\tau$ stopping times, then $\mathbb{E}[M_\tau\mid \mathcal{F_\sigma}]=M_{\sigma\wedge\tau}$
As hint I have that it is convenient to use the martingale $(M_t-M_{t\wedge \tau})_{t\in T}$ and stop it with respect to $\sigma$.
So by considering the predictable function $1_{t\leq \sigma}\in\mathcal{F}_{t-1}$ I consider the stochastic integral $(V\cdot M)_t= \int_0^t 1_{s\leq\sigma}d(M_s-M_{s\wedge \tau})=(M_{t\wedge \sigma}-M_{t\wedge\tau\wedge \sigma})$ which is a martingale. But now I am confused since I have that $\mathbb{E}[M_\tau\mid\mathcal{F}_\sigma]=M_{\sigma\wedge\tau}$. Am I doing all right here or am I doing some mistake? Thanks