I'm currently reading through the following theorem.
Theorem: Let $p$ be a hyperbolic fixed point with $|f'(p)|<1$. Then there is an open interval $U$ about $p$ such that if $x\in U$, then $$\lim_{n\to\infty}f^n(x)=p.$$
With a part of the proof (I only typed up to the point that I got stuck):
Proof: Since $f$ be $C^1$, there is $\epsilon<0$ such that $|f'(x)|<A<1$ for $x\in [p-\epsilon, p+\epsilon]$. By the mean value theorem $$|f(x)-p|=|f(x)-f(p)|\leq A|x-p|<|x-p|\leq \epsilon.$$ ...
Why do we have $|f(x)-f(p)|\leq A|x-p|$ and not $|f(x)-f(p)|< A|x-p|$? Since $|f'(x)|<A$.