Let $[a,b]\subseteq \mathbb{R}$ a non-degenerate interval and $f,g:[a,b]\rightarrow \mathbb{R}$ two continuous functions, differentiable at $(a,b)$ such that |f'(x)| \leq g'(x), \forall x\in(a,b). Prove that $|f(a) - f(b)|\leq g(b) - g(a)$.
I've already proven that |f'(x)| < g'(x) implies $|f(a) - f(b)|\leq g(b) - g(a)$. But I'm having some trouble to show the other one, with the weaker hypothesis.