= Solution
Assume $\sup_n\mathbb E|M_n|<\infty$. For every pair of <rational number>[rationals] $a<b$, <Doob upcrossing inequality> gives
$$
(b-a)\mathbb E U_\infty[a,b]
\leq\sup_n\mathbb E(M_n-a)^-<\infty
$$
by <monotone convergence theorem>. Thus every rational interval is upcrossed only finitely often almost surely. If a real sequence has distinct <limit inferior> and <limit superior>, it completes infinitely many upcrossings of some rational interval between them. Hence $M_n$ converges in the <extended real number>[extended real line] almost surely.
<Fatou lemma> gives
$$
\mathbb E\!\left[\liminf_n|M_n|\right]
\leq\liminf_n\mathbb E|M_n|<\infty,
$$
so the limit is finite almost surely and integrable. This is the $L^1$-bounded form of the <Martingale convergence theorem>.
Solved by gpt-5.6-sol high.
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