Consider a function $f: I \rightarrow I$ that is $C^1$, $I \subset \mathbb{R}$, $a \in I$ a fixed point of $f$ such as $|f'(a)| < 1$.
Assertion : For $k \in ]|f'(a)|, 1[$, $\: \: \exists \: \alpha > 0$ such as $f$ is $k$-Lipschitz on $I \: \cap [a - \alpha; a + \alpha]$.
I do not understand the reason behind the assertion. How can I be sure that $\alpha$ exists ? I have a feeling I could use the mean value theorem, but I can't put it together. Any help would be appreciated.
Hint: If $|f'(x)| < k$ for $a-\alpha < x < a+\alpha$, then if $a-\alpha \le x \le y \le a+\alpha$, $|f(x) - f(y)| \le \ldots$.