= Solution
The radial <null geodesics> satisfy
$$
0=dv(-f\,dv+2\,dr).
$$
The ingoing family has $v=\text{constant}$, with $r$ decreasing toward the future; its tangent $-\partial_r$ is affinely parametrized because $\Gamma^a{}_{rr}=0$. The outgoing family has
$$
\boxed{\frac{dr}{dv}=\frac f2.}
$$
Away from the <Schwarzschild event horizon>, integration gives $v=2r_*+\text{constant}$. These are <null geodesics> up to reparametrization: a null direction in the two-dimensional radial geometry is automatically <pregeodesic>, and the angular <geodesic equations> are satisfied by constant angles. On the <Schwarzschild event horizon> the outgoing family instead has $r=2M$, with tangent proportional to $\partial_v$.
In a <Finkelstein diagram>, plot $T=v-r$ vertically and $r$ horizontally. Ingoing rays obey $T+r=\text{constant}$; outgoing rays obey
$$
T=r+4M\log\left|\frac r{2M}-1\right|+\text{constant}.
$$
Outside the <Schwarzschild event horizon>, outgoing rays increase $r$; on it they remain at $r=2M$; inside it they decrease $r$ even though $v$ increases. Thus both radial future <null directions> point toward smaller $r$ inside the <black hole>. Every future <timelike direction> lies between these <null directions>, so it also moves toward smaller $r$ there.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-311-finkelstein.png]
{title=Radial light rays in ingoing Eddington-Finkelstein coordinates across the Schwarzschild horizon}
{height=480}
Arrows show future propagation. The vertical red line is the <Schwarzschild event horizon>; the black boundary at $r=0$ is the <Schwarzschild singularity>.
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