If I'm Trying to prove $M_n=2^{n}e^{-X_n}$ is Martingale (I've already proven Property Two ie. $E[M_{n+1}\mid F_n]=M_n$.
With Point 1:
$E[|2^{n}e^{-X_n}|]$ If I say $|2^{n}|e^{-X_0}E[|e^{-Y_1}|]< \infty$ .
If the $X_0$ is finite, but $2^n$ isn't wouldn't it not be a MG (but the textbook says it would be). Not sure where to go from here
In point 1, you have to show that for each fixed $n$, $\mathbb E\lvert M_n\rvert$ is finite. You do not have to prove that $\sup_{n\geqslant 1}\mathbb E\lvert M_n\rvert$ is finite (in some cases, it might not be finite).