Help with Theorem - Local Contraction Mapping
I am having some problems understanding the proof of the following 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 .
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 .