Answer
The proof is below.
Work Step by Step
We know that their velocity is equal to $v=\frac{T}{2\pi r}$. We know there is a centripetal force due to gravity, so it follows:
$\frac{Mv^2}{r}=\frac {GM^2}{r^2}$
Using these relationships, we find:
$\frac{r}{GM^2}=\frac{T^2}{m4\pi^2 r^2}$
This simplifies to:
$T^2 = \frac{4\pi^2r^3}{GM}$
