I have an exercise to prove something in contraction mapping
I think it is easy but I'm not familiar with the subject.
It is REALLY urgent
Thanks
The first problem is quite straightforward: Let $\displaystyle \delta\in(|x_{n+1}-x_n|,\epsilon)$, and notice that $\displaystyle |x_{n+1}-x_{n+2}|<\gamma\delta$. (This is by definition of the sequence $\displaystyle \langle x_k\rangle$.) In fact, it's easy to see that by induction $\displaystyle |x_{n+i}-x_{n+i+1}|<\gamma^i\delta$ for $\displaystyle i=0,1,2,\cdots$. So by the triangle inequality
$\displaystyle |x_n-x_r|\leq \sum_{i=0}^{r-1}|x_{n+i}-x_{n+i+1}|<\sum_{i=0}^{r-1}\gamma^i\delta$.
Letting $\displaystyle r\to\infty$ gives us $\displaystyle |x_n-x^*|\leq\sum_{i=0}^\infty\gamma^i\delta=\frac{ \delta }{1-\gamma}<\frac{\epsilon}{1-\gamma}$.
(Here we have used the identity of a geometric series.)
I have no idea how to do the second problem, sorry.