Proving a stopping time Let $$\mathcal F_s = \{ A : A\cap \{S \leq t\} \in \mathcal F_t, \forall t \geq 0\}$$
where $\left(\mathcal F_t\right)_{t\geqslant 0}$ is a right continuous filtration.
Let $S$ be a stopping time, let $A \in\mathcal F_s$ and let $R=S$ on $A$ and $R = \infty $ on $A^c$. Show that $R$ is a stopping time. 
I'm really lost on this question, it's question 7.3.4 in Rick Durrett's Probability Theory book. 

Fix $t\ge0$. Then
\begin{align*}
\{R \le t\} &= (\{R \le t\} \cap A) \cup (\{R \le t\} \cap A^c)\\
&= (\{S \le t\} \cap A) \cup \emptyset\\
&= \{S \le t\} \cap A \in \mathcal{F}_t,
\end{align*}
since $A\in\mathcal{F}_S$. Since $t$ was arbitrary, this shows $R$ is a stopping time.
