Help with Theorem - Local Contraction Mapping

Hi,

I am having some problems understanding the proof of the following theorem.

**Theorem: **

Suppose that g is a real-valued function, defined and continuous on a bounded closed interval of the real line, and assume that for all .

Let be a fixed point of , and assume that has a continuous derivative in some neighborhood of with . Then the sequence defined by , converges to as , provided that is sufficiently close to .

**Proof: **

By hypothesis, there exists such that is continuous in the interval

. Since we can find a smaller interval

, where , such that in this interval, with .

To do so, take and then choose such that,

for all in ; this is possible since is continuous at .

I will stop there as already I am not sure what's going on.

Why should I take ?

By the way, in this book L is used to denote the "contraction factor" such that;

for all .

Thanks.