On page 10 of Rudin's Principles of Mathematical Analysis, Rudin makes a claim that the identity
$b^{n} - a^{n} = (b - a)(b^{n-1} + b^{n-2}a + \cdots + a^{n-1})$
yields the inequality $b^{n} - a^{n} < (b - a)nb^{n-1}$ when $0 < a < b$, for real $a$ and $b$, and natural $n$.
Is this result obvious or does it actually require proving? Rudin gives no explanation for why it's true and I failed at finding any obvious reasons for it being true.