Answer
The proof is below.
Work Step by Step
We know equation 8.5:
$U=GMm(\frac{1}{r_1}-\frac{1}{r_2})$
We must show that $U=mg\Delta r$, so we find:
$GMm(\frac{1}{r_1}-\frac{1}{r_2})=mg\Delta r$
$GM(\frac{1}{r_1}-\frac{1}{r_2})=g\Delta r$
We know from the book the $\Delta r = r_1-r_2$, so we find:
$GM(\frac{1}{r_1}-\frac{1}{r_2})=g( r_1-r_2)$
$g( r_1-r_2)=g( r_1-r_2)$
