I have been wondering about the following equality in the textbook by Liggett. I put a red circle at the position where my question is. They use the theorem that $B_t^2-t$ is a martingale and the martingale stopping theorem to argue that $B_{t \wedge n}^2-(\tau \wedge n)$ is a martingale and from this they derive the equality of expectation values, but I don't see how this follows.
Does anybody have an idea?
Theorem 1.102. If $\tau$ is a stopping time with $E_\tau<\infty$, then
(a) $EB(\tau)=0;$
(b) $EB^2(\tau)=E_\tau;$
(c) $E_\tau^2\le4EB^4(\tau).$Proof. It is easier to prove the first two parts together. By Theorems 1.95(b) and 1.93, $B^2(\tau\wedge n)-\tau\wedge n$ is a martingale. Therefore $$EB^2(\tau\wedge n)\require{enclose}\color{red}{\enclose{circle}=}E(\tau\wedge n)\le E_\tau<\infty\tag{1.32}$$
$T=\min(t,n)$ is a bounded stopping time. By Optional Stopping, the expectation is $$E(B^2_T -T) = E(B^2_0 -0) = 0$$ Conclude by linearity of expectation.