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\newcommand{\ds}{\displaystyle}
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\begin{document}

{\bf Question}

Write down the proof that if the fixed point $p$ for $f:\ {\bf R}
\longrightarrow {\bf R}$ has $|f'(p)|>1$ then $p$ is a source.
\medskip

{\bf Answer}

Choose $c$ with $|f'(p)|>c>1$. Since by definition $f'(p)=\lim_{x
\to p} \ds\frac{f(x)-f(p)}{x-p}$ there is some $N_{\epsilon}(p)$
with $\left|\ds\frac{f(x)-p}{x-p}\right|>c$ for all $x \leq
N_{\epsilon}(p),\ x \ne p$, i.e. $|f(x)-p|>c|x-p|$ for all $x \in
N_{\epsilon}(p)$. Thus if $x,f(x),...,f^{n-1}(x)$ all $\in
N_{\epsilon}(p)$ we have $|f^n(x)-p|>c^n|x-p|$ so eventually
$f^n(x) \not\in N_{\epsilon}(p)$.

Therefore $p$ is a source.
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