I'm working on a proof in real analysis. Here is the body of the exercise:
Let $g: A \rightarrow \mathbb{R}$ be a differentiable function where $g'$ is continuous and $g'(x) < 1$ on a closed interval $I$. Prove that there exists a constant $c$ such that for all $x,y \in I$, $$|f(x)-f(y)| \leq c|x-y|. $$
I think I have done a similar proof before: I think it all rides on how you define your $\epsilon$ and $\delta$. I would appreciate help on constructing this solution. Thank you.
Think of the fundamental theorem of calculus i.e. $$ f(x) - f (y) = \int_y^x f'(x) dx$$ and use some property of the integral.