Contraction Mapping Theorem / Simple iteration

So, I am reading in a introductory numerical analysis book, but I believe my question fits into the calculus forums.

First, the definition of a simple iteration as given in my book:

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

* is called a ***simple iteration**.

__Contraction Mapping Theorem:__

*Let g be as in the definition above. Suppose further, that g is a contraction on . Then, has a unique fixed point in the interval . Moreover the sequence defined above converges to as for any starting value in .*

First of all, I don't really understand the simple iteration. I see that because for all , then there exists a number such that . It would be great if someone had the time to explain this in more detail.

As for the contraction mapping theorem, I understand how has a unique fixed point in the interval . However, I do not understand how it implies that the recursive sequence converges to as .

The proof of this is given in the book and is as follows;

we then deduce by induction that

I do not understand how they come to that last part by induction.

Any help is appreciated. Thanks!