= Solution
{c}
The <supermartingale> property and linearity of <conditional expectation> give, for $n\geq1$,
$$
G_n=\mathbb E(X_n\mid\mathcal F_{n-1})-X_{n-1}\leq0.
$$
Each $G_n$ is integrable and $\mathcal F_{n-1}$-measurable, while $G_0=0$. Thus $Z_n$ is integrable and
$$
\boxed{Z_{n+1}-Z_n=G_{n+1}\leq0.}
$$
Moreover, all terms $G_j$, $j\leq n+1$, are $\mathcal F_n$-measurable, so $Z_{n+1}\in m\mathcal F_n$. This is predictability of the decreasing component in the <Doob decomposition theorem>.
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