= Solution
For $1<p<\infty$, the <Lp martingale convergence theorem> states that a <martingale> with $\sup_n\mathbb E|M_n|^p<\infty$ has a limit $M_\infty\in L^p$ such that
$$
\boxed{M_n\to M_\infty\text{ almost surely},\qquad
\|M_n-M_\infty\|_p\to0.}
$$
Moreover, $M_n=\mathbb E[M_\infty\mid\mathcal F_n]$ and $\mathbb E|M_\infty|^p\leq\sup_n\mathbb E|M_n|^p$. To see why $p>1$ matters, the <Doob Lp maximal inequality> gives an integrable dominating variable $(\sup_n|M_n|)^p$. The <Martingale convergence theorem> first supplies the almost-sure limit, then the <dominated convergence theorem> supplies convergence in the <Lp norm>. This maximal estimate is unavailable at $p=1$ in the required form.
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