Let $\tau$ be a stopping time and $(M_n)_{n \in \mathbb{N}_0}$ be a martingale with $\mathbb{E}(M_n^2)<\infty$ for any $n \in \mathbb{N}_0$. Show that, if $\langle M \rangle_{\tau} = 0$ (where it means the quadratic variation) a.s. then $$\mathbb{P}(M_{\tau \wedge n} = M_0 \, \, \text{for any $n \in \mathbb{N}_0$})=1.$$
I cannot understand the point, in fact $$\mathbb{P}(M_{\tau \wedge n} = M_0 \, \, \text{for any $n \in \mathbb{N}_0$})=1$$ I think it means that the stopping time is $\tau=0$ but it makes no sense. Could someone help me to resolve this exercise?