1. ## Contraction Mapping.

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

2. ## Re: Contraction Mapping.

The first problem is quite straightforward: Let $\delta\in(|x_{n+1}-x_n|,\epsilon)$, and notice that $|x_{n+1}-x_{n+2}|<\gamma\delta$. (This is by definition of the sequence $\langle x_k\rangle$.) In fact, it's easy to see that by induction $|x_{n+i}-x_{n+i+1}|<\gamma^i\delta$ for $i=0,1,2,\cdots$. So by the triangle inequality

$|x_n-x_r|\leq \sum_{i=0}^{r-1}|x_{n+i}-x_{n+i+1}|<\sum_{i=0}^{r-1}\gamma^i\delta$.

Letting $r\to\infty$ gives us $|x_n-x^*|\leq\sum_{i=0}^\infty\gamma^i\delta=\frac{ \delta }{1-\gamma}<\frac{\epsilon}{1-\gamma}$.

(Here we have used the identity of a geometric series.)

I have no idea how to do the second problem, sorry.

3. ## Re: Contraction Mapping.

Thank you very much, last night I tried to solve it and I did as shown in the attached pdf file.
I would like to know if there is any comment

4. ## Re: Contraction Mapping.

Looks good to me! I wouldn't have thought to use the Banach fixed point theorem (since I had never heard of it until just now). But it works just fine.