Probability with Martingales:
I have a different attempt in mind, but I'm guessing it's wrong because if it were right, the book would've used it.
It seems that we must show that
$$A_{S_k \wedge n} \ \in \ m \mathscr F_{n-1}$$
Case 1: $$n \le S_k$$
$$A_{S_k \wedge n} = A_{n} \ \in \ m \mathscr F_{n-1}$$
Case 2: $$n > S_k \iff n-1 \ge S_k$$
$$A_{S_k \wedge n} = A_{S_k} \ \in \ m \mathscr F_{S_k}$$
Since $\mathscr F_{S_k} \subseteq \mathscr F_{n-1}$,
$$A_{S_k} \ \in \ m \mathscr F_{n-1}$$
QED
Did I go wrong somewhere? Are we forbidden from using a stopping time for a $\sigma$-algebra?