= Solution
A spherical <trapped surface> has both future <null expansions> negative. Thus the initial sphere has $r_{,U}<0$ and $r_{,V}<0$. The result of the previous part already gives $r_{,U}<0$ at every regular future sphere in $D^+(\Sigma)$.
Apply the same <Null Raychaudhuri equation> to $k=f^{-1}\partial_V$. The exchanged-coordinate <double-null focusing identity> is
$$
\partial_V(f^{-1}r_{,V})=-\frac r{2f}R_{VV}=-\frac{4\pi r}{f}T_{VV}\leq0.
$$
Along $U=U_0$, increasing $V$ is future-directed. Therefore
$$
f^{-1}r_{,V}(U_0,V_1)\leq f^{-1}r_{,V}(U_0,V_0)<0.
$$
Both <null expansions> remain negative, proving \b[<persistence of spherical trapping> along this future null direction]. As usual, the conclusion concerns every $V_1>V_0$ for which the sphere exists in the regular chart in $D^+(\Sigma)$; the inequality does not assert existence of such spheres beyond a focusing singularity.
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