Prove Doob's using a certain Lemma

I am to prove Doob's (d) in the red box below:

What I tried:

Since $T < \infty$ a.s., we have

$$E[X_T] = E[\lim X_{T \wedge n}].$$

By Fatou's Lemma, we have

$$E\left[\lim X_{T \wedge n}\right] = E\left[\liminf X_{T \wedge n}\right] \stackrel{\mbox{Fatou's}}{\le} \liminf E\left[X_{T \wedge n}\right] \le \liminf E[X_0] = E[X_0].$$

Is that right?

1 Answer

I think you just need to change your second-to-last $=$ in a $\leq$ as you have a super martingale and not a martingale. Other than that, I think you are right.