= Solution
Put $C=\sup_p\mathbb E|X_p|<\infty$. Since $x\mapsto x^+$ is an increasing <convex function>, the <conditional Jensen inequality> and the <submartingale> property give
$$
\mathbb E[X_{p+1}^+\mid\mathcal F_p]\geq\bigl(\mathbb E[X_{p+1}\mid\mathcal F_p]\bigr)^+\geq X_p^+.
$$
For fixed $n$ and $p\geq n$, the <tower property of conditional expectation> therefore implies
$$
\mathbb E[X_{p+1}^+\mid\mathcal F_n]\geq\mathbb E[X_p^+\mid\mathcal F_n].
$$
Choose versions for this countable family so that all these inequalities hold outside one null set. Its nonnegative increasing limit $M_n$ is $\mathcal F_n$-measurable. The <monotone convergence theorem> shows that
$$
\mathbb EM_n=\lim_{p\to\infty}\mathbb EX_p^+\leq C.
$$
In particular the limit is finite <almost surely>. Thus \b[the increasing conditional means converge to an integrable nonnegative $M_n$], rather than merely to a possibly infinite extended value. This is the construction of the <positive martingale majorant of an L1-bounded submartingale>.
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