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 $\kappa <1$
such that $|e_k|$$\leq C\kappa^k $
d.)then deduce by showing all steps that $\lim_{k \to \infty} x_k = 1/2 $
Obviously for part C i have to use some kind of theorem or formula but i am unsure whether t means something like banach caccioppoli or a basic analysis theorem. I also know that the error term is generally defined as $e_k:=\hat{x}-x_k$ but i am unsure how i can apply this to prove/show the statement. Please help! Many thanks :)