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**Pinkk** Let $\displaystyle f$ be differentiable on $\displaystyle \mathbb{R}$ with $\displaystyle a=\sup \{|f'(x)|; x\in \mathbb{R}\}< 1$. Select $\displaystyle s_{0}\in \mathbb{R}$ and define $\displaystyle s_{n}=f(s_{n-1})$ for $\displaystyle n\ge 1$. Prove that $\displaystyle (s_{n})$ is a convergent sequence. *Hint: To show $\displaystyle (s_{n})$ is Cauchy, first show that $\displaystyle |s_{n+1} - s_{n}| \le a|s_{n} - s_{n-1}|$ for $\displaystyle n\ge 1$.*

Okay, so it's pretty clear that I have to use the mean value theorem and so there exists $\displaystyle c\in \mathbb{R}$ such that $\displaystyle \frac{f(b)-f(a)}{b-a}=f'(c)$ for $\displaystyle a,b\in \mathbb{R}$. Now it seems that I should just replace $\displaystyle a,b$ with $\displaystyle s_{n},s_{n-1}$ and I will arrive at the hint. However, how can I guarantee that for any $\displaystyle n$, $\displaystyle s_{n}\ne s_{n-1}$? Is there something I am missing that guarantees this, or is my actual approach incorrect so far? And even if I arrive at the hint, how would I proceed exactly? Thanks.