Inequality real analysis

$$|a^n-b^n|=|a-b|\left|\sum_{k=0}^{n-1}a^kb^{n-k-1}\right|\leq |a-b|M^{n-1}n$$

Using the mean value theorem for the function $f(x) = x^n$ we get that there exists some $\theta \in \langle a,b\rangle$ such that $$|a^n - b^n| = |f(a) - f(b)| = |f'(\theta)||a-b| = n|\theta|^{n-1}|a-b| \le nM^{n-1}|a-b|$$ since $|\theta| \le \max\{|a|, |b|\} = M$.