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**nqramjets** This one has been stumping me for quite some time know...

Given $\displaystyle f:[a,b]\rightarrow[a,b]$ continuous on $\displaystyle [a,b]$ and differentiable on $\displaystyle (a,b)$ such that there exists an $\displaystyle \alpha\in(0,1)$ so that $\displaystyle |f'(x)|<\alpha$ for all $\displaystyle x\in(a,b)$, show that there exists a unique point $\displaystyle c\in[a,b]$ such that $\displaystyle f(c)=c$.

I can show that if there is a point, then it is unique, but showing the point exists is challenging.

Hint: Pick $\displaystyle x_0\in[a,b]$ and let $\displaystyle x_{k+1}=f(x_k)$.

I have used repeated applications of the mean value theorem to achieve the following inequality

$\displaystyle |x_{k+1}-x_k|\leq\alpha^k|x_1-x_0|$

Unfortunately, this is not Cauchy necessarily, so I am very stuck. Any help would be greatly appreciated!