I was trying to solve Problem 3.23 (b) in Chapter 1 of Karatzas and Shreve
where we need to establish the optional sampling theorem(in the picture below)for a right continuous submartingale and optional time $S\leq T$ under the assumption that $T \leq a$ for some $a>0$
.
I tried to adapt the proof in the discrete case but since we don't have bounded stopping times and the stochastic processes $X$ is not given to be closed by a last element $X_{\infty}$, I cannot apply the theorem. In particular as you can see below theorem 3.22 uses an backward submartingale argument with $X_{S_n}$ being a $\mathcal{F}_{S_n}$ backward martingale(which is due to the fact that in Theorem 3.22 we know that X is closed)
Can somebody show me how does boundedness of the stopping time help in establishing the optional sampling theorem in this case? If we use the same approximating sequence of stopping times $S_n$ and $T_n$ as used in Theorem 3.22, we cannot conclude that they are bounded since they approximate $S$ and $T$ from above!