Proving Banach's fixed point theorem

Question

The hint tells me how to proceed but I am stuck.

I define the sequence ${z_n}$ as is stated in the hint,

First off, I want to prove that $|z_{n+k} - z_n| < \epsilon$

I add $z_{n+k-1}$ and subtract it (adding zero) so I get

$|z_{n+k} + z_{n+k-1} - z_{n+k-1} - z_n|$

Can I use the triangle inequality on this? When do I invoke the contraction ?

Answer 1

You want to show that $|z_{n+k}-z_n|<\epsilon$ for $n$ big enough. First you note that $$ |z_{m+1}-z_m|=|T(z_m)-T(z_{m-1})|<c\,|z_m-z_{m-1}|<c^2\,|z_{m-1}-z_{m-2}|<\cdots<c^m\,|z_1-z_0|. $$

Then you have $$ |z_{n+k}-z_n|=\left|\sum_{j=0}^{k-1}z_{n+j+1}-z_{n+j}\right|\leq\sum_{j=0}^{k-1}|z_{n+j+1}-z_{n+j}|<\sum_{j=0}^{k-1}c^{n+j}\,|z_1-z_0|\\ =c^n\,|z_1-z_0|\,\sum_{j=0}^{k-1}c^j=c^n|z_1-z_0|\,\frac{1-c^k}{1-c}\leq\,c^n\,\frac{|z_1-z_0|}{1-c} $$