= Solution
For an <asymptotically flat spacetime> with a chosen exterior component of <future null infinity>, the <black-hole region> is
$$
\boxed{\mathcal B=\mathcal M\setminus J^-(\mathscr I^+),}
$$
namely the events from which no future-directed <causal curve> can reach that infinity. Here $\mathscr I^+$ is the infinity of the original $r>r_+$ exterior.
Let a future causal tangent in the <ingoing Eddington-Finkelstein coordinates> have components $(\dot v,\dot r,\dot\Omega)$. Its inner product with the future null field $n=-\partial_r$ is $-\dot v$, so $\dot v\geq0$. The causal inequality is
$$
-f\dot v^2+2\dot v\dot r+r^2|\dot\Omega|^2\leq0.
$$
If $\dot v>0$, this implies
$$
\dot r\leq\frac f2\dot v-\frac{r^2|\dot\Omega|^2}{2\dot v}.
$$
In the band $r_-<r<r_+$, $f<0$, so $\dot r<0$. If $\dot v=0$, the causal inequality forces $\dot\Omega=0$, and future direction means the remaining radial tangent is a nonnegative multiple of $n$, again giving $\dot r\leq0$. This is <causal trapping between two spherical horizons>: \b[$r$ is nonincreasing along every future causal curve in the between-horizon band]. No event there can escape outward. An event with $0<r\leq r_-$ would also have to cross this band outward to reach the original exterior, which is impossible. Thus all $0<r<r_+$ events in this extension lie in the <black-hole region>.
Conversely, from any $r>r_+$ point, the outgoing radial null ray $u=\mathrm{constant}$ satisfies $dr/dv=f/2>0$. It reaches arbitrarily large $r$ and then the original <future null infinity>. Therefore \b[the exterior does not intersect the <black-hole region>], and $r=r_+$ is its <event horizon>. This conclusion is relative to the selected asymptotic end; the maximal charged-black-hole extension can have other asymptotic ends.
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