I'm trying to follow a proof and am having difficulty understanding it. The proof is in Williams Probability with Martingales, where 10.7 and 10.9 are used to show the theorem on page 99.
Essentially, let $X_t$ be a discrete-time martingale, $\tau$ a stopping time, and suppose I've shown $Y_t=X_{\tau\land t}-X_0$ is a martingale. Williams immediately concludes that $\mathbb EY_t=0$, but I'm not sure how he attained that result. It seems to me that what I specifically know is $\mathbb E_{\mathcal G_{t-1}}Y_{t}=Y_{t-1}$, or that, via the tower rule, $\mathbb E_{\mathcal G_0}Y_t=Y_0=0$ but then I'd still be trapped with $\mathbb E_{\mathcal G_0}X_{\tau\land t}=X_0$ after simplifying.
What's the obvious thing I'm missing?
you are almost there… you already have $\mathbb E_{\mathcal G_0}Y_t=0$. Taking expectations on both sides gives you $E[Y_t] = 0$ due to the fact, that for conditional expectation holds $E[E_\mathcal{G}[X]] = E[X]$