Probability with Martingales:
It looks like we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where
$A = \{A_n\}_{n \in \mathbb N}$ is an $\{\mathscr F_n\}_{n \in \mathbb N}$-adapted process which is previsible meaning it is $\{\mathscr F_{n-1}\}_{n \in \mathbb N}$ adapted.
To prove $S_k$ is a stopping time, we have to prove that
$$\{S_k = m\} \in \mathscr F_m \ \forall m \in \mathbb N$$
Now
$$\{S_k = m\} = \{\inf\{n \in \mathbb Z^{+} \ : \ A_{n+1} > k\} = m\}$$
I'm guessing $$\{\inf\{n \in \mathbb Z^{+} \ : \ A_{n+1} > k\} = m\} =$$
$$\bigcap_{n=0}^{m} \{A_n \le k\} \cap \{A_{m+1} > k\} \in \mathscr F_m$$
$$\because \bigcap_{n=0}^{m} \{A_n \le k\} \in \mathscr F_m \ \text{and} \ \{A_{m+1} > k\} \in \mathscr F_m$$
Is that right?
Also, how should I interpret $S_k$? The first time $A$ exceeds k plus 1?
$\{S(k)\le m\} =\{A_{m+1}>k\}\in\mathscr F_m$, because $A$ is previsible.