Ok so I'm trying to prove two results for the standard American put option with pricing function: $$P^A(t,S(t))=\sup_{\tau\in (t,T)} \mathbb{E}_{\mathbb{Q}}\left [ e^{-r(\tau-t)}(K-S(\tau))^{+} | \mathcal{F}_t\right ]$$
Where $K$ is the strike, $T$ the expiry, and we are taking the supremum over all stopping times in the interval $(t,T)$. Here are the proofs I'm trying to do:
$\textbf{a)}\;$Denote by $\bar{P}^A(t,S(t))=e^{-rt}P^A(t,S(t))$ the discounted pricing function of the American put option. Show that $\bar{P}^A(t,S(t))$ is a supermartingale under the measure $\mathbb{Q}$.
$\textbf{b)}\;$ Denote $\tau^*(t)$ as the exercise time of the option after $t$, show that $\bar{P}^A(t,S(t))$ is a martingale under $\mathbb{Q}$ in the time interval $[t,\tau^*(t)]$, i.e., for any stopping time $t\leq \tau \leq \tau^*(t)$, $\mathbb{E}_{\mathbb{Q}}[\bar{P}^A(\tau,S(\tau))|\mathcal{F}_t]=\bar{P}^A(t,S(t))$
I believe I have a solid proof for $\textbf{a}$ however I am pretty stuck at $\textbf{b}$. Here's my proof for $\textbf{a}$.
Consider $t'<t$, then conditioning on this we have: $$\mathbb{E}_\mathbb{Q}[\bar{P}^A(t,S(t))|\mathcal{F}_{t'}]=\mathbb{E}_\mathbb{Q}\left [e^{-rt}\sup_{\tau\in (t,T)} \mathbb{E}_{\mathbb{Q}}\left [ e^{-r(\tau-t)}(K-S(\tau))^{+}\rvert \mathcal{F_{t}}\right ]\rvert \mathcal{F_{t'}}\right ]$$ $$\leq \sup_{\tau\in (t',T)}\mathbb{E}_\mathbb{Q}\left [ \mathbb{E}_{\mathbb{Q}}\left [ e^{-r\tau}(K-S(\tau))^{+}\rvert \mathcal{F_{t}}\right ]\rvert \mathcal{F_{t'}}\right ]$$ Where we used the expectation conditioned on $\mathcal{F_{t'}}$ to change the $\tau$ in the supremum and then took it out to create the inequality. Now by tower property we have $$=\sup_{\tau\in (t',T)}\mathbb{E}_\mathbb{Q}\left [ e^{-rt'}e^{-r(\tau-t')}(K-S(\tau))^{+}\rvert \mathcal{F_{t'}}\right ]$$ and hence taking $e^{-rt'}$ out the result follows.
Now for part $\textbf{b}$ I'm not even sure where to start. The question seems very ambiguous, is it asking to consider the same case but restricted to the interval $(t, \tau^*(t))$? Or is it asking to consider this interval but the pricing function is only evaluated at the stopping times between $t$ and $\tau^*(t)$? I feel like if it was the latter we could apply the optional stopping theorem, however that would imply that it is again a supermartingale. Any pointers to get me started would be greatly appreciated.