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It is an exercise from Kincaid and Cheneys's book.

How can we show that $x_{n+1}=f(x_{n})$ will converge if $|f'(x)|\leq\lambda<1$ on the interval $I=[x_{0}-\rho, x_{0}+\rho]$ where $\rho = \frac{|f(x_{0})-x_{0}|}{1-\lambda}$?

My idea is to show that $f$ maps $I$ to itself. Then Contractive mapping theorem guarantee that the sequence will converge.

But I don't see a way to show it.

Any idea and help would appreciated?

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There you go ; you had the right idea! This is how to show it : $$ |f(x) - x_0| \le |f(x) - f(x_0)| + |f(x_0) - x_0| = |f'(c)||x-x_0| + \rho(1-\lambda) \le \lambda \rho + \rho(1-\lambda) = \rho. $$ Therefore $f$ maps $I$ to itself. I used Taylor's theorem to get $c$ between $x_0$ and $x$. Afterwards you can use the contractive mapping theorem (which by the way is really not that hard to prove if you've never seen a proof of it).

Hope that helps,

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