Can someone give an insight on the following problem? I'm not sure how to start the problem. It's a practice problem for "mean value theorem" and "Taylor's Theorem" so I'm assuming they might be necessary for the proof. Thanks!
Let $f: \mathbb{R} \to \mathbb{R}$ be a function. Suppose that $f$ is differentiable, that $f(0)=1$, and that $|f'(x)| \leq 1$ for all $x \in \mathbb{R}$. Prove that $|f(x)| \leq |x|+1$ for all $x \in \mathbb{R}$.