= Solution
The <almost sure supermartingale convergence theorem> says that a supermartingale whose negative parts have uniformly bounded expectations converges almost surely to a finite integrable limit. In particular, every nonnegative supermartingale converges almost surely.
Here the process is uniformly bounded, say $0\leq X_n\leq C$. Let $X_\infty=\lim_nX_n$, whose existence follows from the theorem. The <dominated convergence theorem> then gives
$$
\mathbb E[X_\infty]=\lim_{n\to\infty}\mathbb E[X_n].
$$
The limit on the right also exists directly because the expectations of a <supermartingale> form a decreasing sequence.
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