Prove that the sequence generated by the iteration $x_{n+1} = F(x_n)$ will converge if $|F^{'}(x)| \le \lambda < 1$ on the interval $[x_0 -p , x_0 + p]$ where $p = \frac{|F(x_0) - x_0|}{1- \lambda} $
My attempt to solve this problem was using the contractive mapping theorem, since maximum value of derivative is less than one in the closed set $[x_0 - , x_0 + p]$ it's a contraction but I still have to prove that the function maps $[x_0 - , x_0 + p]$ into itself and that's where I don't know how to prove.
This is a question from D Kincaid & W Cheney, Numerical Analysis (3ed) Section 3.4
Hint: Prove by induction that $|x_{n+1}-x_n| \le \lambda^{n}|F(x_0)-x_0|$ and $|x_{n+1}-x_0| \le \frac{1-\lambda^{n+1}}{1-\lambda}|F(x_0)-x_0|$