# Fixed Point theory question (Numerical methods)

I have an exam in a previous exam paper which i have no solutions too. I am stuck on the last 2 parts of the question and have been for several days now! Any help much appreciated. Here is the question:

Let $D:=[1/4,3/4]$

and consider the iteration $x_k = 2x_{k-1} (1-x_{k-1})$ with any fixed $x_0 \in D$

Define the residual $r_k:=x_k - x_{k-1}$

and define the error $e_k:=x_k - 1/2$

c.)Argue from first principles that there exists constants $C$,$\kappa >0$ with $\kappa <1$