= Solution
Suppose the closed <trapped surface> $T$ were not wholly in the <black hole>. The stated consequence of <strong asymptotic predictability> then supplies a point $p\in\dot J^+(T)\cap\mathcal I^+$ lying on a future null generator $\gamma$ orthogonal to $T$, with no point conjugate to $T$ before $p$. Both future null expansions at $T$ are negative. The <null focusing theorem> therefore forces the expansion along $\gamma$ to diverge after finite <affine parameter>, producing a <conjugate point> before the generator reaches <future null infinity>. A null geodesic with such a point cannot continue to generate the achronal boundary $\dot J^+(T)$, contradicting the property of $p$. Hence no point of $T$ can communicate with future null infinity, and
$$
\boxed{T\subset B=M\setminus J^-(\mathcal I^+)}.
$$
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