Exercise with a weaker hypothesis 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. 

One solution for such problems is to add in some $\epsilon$:
$g(x) + \epsilon (x-a)$ satisfies your stronger hypothesis for every $\epsilon > 0$, so 
$$|f(b) - f(a)| \leq g(b) - g(a) + \epsilon (b-a).$$
As $\epsilon \to 0$, you obtain the desired result.
