In the definition of a local martingale I have that for a filtered probasbility space $(\Omega, \mathcal{F},P,\mathbb{F})$. A local martingale is an adapted process M, such that there exists a sequence of increasing stopping times such that $\tau_{n}\rightarrow \infty$ a.s., and $M(t\wedge\tau_n)$ is a martingale for each n. This is also the definition given on Wikipedia.
I have a problem showing that if M is a local martingale, and $\tau$ is a stopping time, then $G(t)=M(t\wedge \tau)$ is a local martingale. The proof I have been given seems wrong:
By doobs optional sampling theorem we can show that for each n:
$E[G(t\wedge \tau_n)|\mathcal{F}_{s\wedge \tau}]=E[M(t\wedge \tau\wedge \tau_n)|\mathcal{F}_{s\wedge \tau}]=E[M((t\wedge \tau)\wedge \tau_n)|\mathcal{F}_{s\wedge \tau}]=_\text{optional sampling}M((s\wedge \tau)\wedge \tau_n)=G(s\wedge \tau_n)$.
In the theorem I used the two bounded stopping times $t\wedge \tau$ , and $s \wedge \tau$.
But this only shows that $E[G(t\wedge \tau_n)|\mathcal{F}_{s\wedge \tau}]=G(s\wedge \tau_n)$. But I need that $E[G(t\wedge \tau_n)|\mathcal{F}_{s}]=G(s\wedge \tau_n)$.
Do you see how to finish the proof? I am stuck.