Show that radial worldlines with u = const are outgoing null rays

[so_gr_lo]

Homework Statement

The transformation from Schwarzschild coordinates (t,r,θ,φ) to outgoing Eddington– Finkelstein coordinates (u, r′, θ′, φ′) is given by
u = t−r∗
r′ = r
θ′ = θ
φ′ = φ,
where r∗ satisfies dr∗/dr = (1−2M/r)−1. Use the rule for the coordinate transformation
of metric components to obtain the line element in the (u, r′, θ′, φ′) coordinates.

(b) Show that radial worldlines with u = const are outgoing null rays.

I have managed part a but am not sure how to show how the line element I have is null.

I have used the fact that u = t−r∗ suggests du=0 and $d{\Omega}^2$ = 0 to simplify the line element in (a) and give the result below. Does anyone know how I can show that the result is null?

Relevant Equations

u = t−r∗
r∗ satisfies dr∗/dr = (1−2M/r)−1

$$ ds^2 = -(1-\frac{2M}{r'})(\frac{\frac{r'}{2M} -1+2M}{\frac{r'}{2M}-1})(dr')^2+(1-\frac{2M}{r'})^-1 (dr')^2 $$