= Solution
{c}
The <uniformly integrable martingale convergence theorem> states that a <uniformly integrable> <martingale> $(U_n)$ has an <integrable> terminal <random variable> $U_\infty$ such that $U_n\to U_\infty$ <almost surely> and in the <L1 norm>, and
$$
U_n=\mathbb E[U_\infty\mid\mathcal F_n].
$$
The last identity follows by passing to the <limit of a sequence> in $\mathbb E[U_m\mathbf1_A]=\mathbb E[U_n\mathbf1_A]$ for $A\in\mathcal F_n$ and $m\geq n$, using <convergence in L1>.
Part (b)(iii) verifies <uniform integrability> for $(Z_n)$, so
$$
\boxed{Z_n\longrightarrow Z_\infty\quad\text{almost surely and in }L^1,\qquad
Z_n=\mathbb E[Z_\infty\mid\mathcal F_n].}
$$
Moreover $|Z_\infty|\leq L$ <almost surely>. In this particular problem the common bound also yields <convergence in Lp> for every finite $p\geq1$, including <convergence in L2>: the <dominated convergence theorem> applies to $|Z_n-Z_\infty|^p\leq(2L)^p$. General <uniform integrability> alone does not assert these higher-power conclusions.
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