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?