= Solution
The <almost sure submartingale convergence theorem> states that a discrete-time <submartingale> $Y_n$ satisfying $\sup_n\mathbb E Y_n^+<\infty$ converges <almost surely> to a finite integrable limit. In particular, \b[a martingale with $\sup_n\mathbb E|M_n|<\infty$ converges almost surely to an integrable $M_\infty$], and $\mathbb E|M_\infty|\leq\sup_n\mathbb E|M_n|$ by <Fatou's lemma>. This is the <martingale convergence theorem>. <Uniform integrability> gives the stronger conclusion of <L1 convergence>; it is not implied by bounded first moments alone.
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