# fixed point in the infinite, involving derivates, and composite of a $C^1$ function

Hello this problem it´s looks easy, but I can´t do it. If you can give me some hint to do it )= It´s says this Let $f:\left[ {a,b} \right] \to \left[ {a,b} \right]$ be $C^1$. Let $p$ a fixed point of $f$ such that $| f´(p) | < 1$ Prove that there exist $\delta >0$ such that for every $x \in \left( {p - \delta ,p + \delta } \right)\Rightarrow \,\mathop {\lim }\limits_{n \to \infty } f^n \left( x \right) = p$.

Where $f^n \left( x \right)$ denotes the composite of functions , $n$ times

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Hint: if $|f'(x)| < 1-\epsilon$ for $|x - p| < \delta$, what can you say about the relation of $|f^{n+1}(x) - p|$ to $|f^n(x) - p|$?
it´s only in "p" the inequality $| f´(p ) | < 1$ where "p" is the fixed point –  August Oct 9 '11 at 17:59
Yes I did it, i did not consider the hypothesis of being $C^1$ , with this the delta required for example can be taked in such a way , that $f´(x ) < 1- epsilon$ as you said, thanks! –  August Oct 9 '11 at 18:23