Let I be an interval of the real line and be a

continuous function from I onto itself. Apply it repeatedly and the process is

kind of a feedback loop. x results in an output y=f(x).

After a single period, the new value of x is f(x), after 2 periods it's , and on and on.

The sequence: is called theorbitof x.

Let's see. Let I be the set of non negative reals and define:

for every x in I. We have the iterations:

The orbit of 1:

This sequence is increasing since every term after the second is gotten from

the previous one by replacing the final b with

Therefore, the orbit(bounded and

monotonic), is convergent.

If we let its limit be L, then:

shows that, as ,

Therefore, L is a fixed point of f. A solution of x=f(x).

With some basic algebra we see

I hope I didn't delve into too much. I just wanted to explain best I could.

I wanted to show more about its convergence, but I'm tired of fighting with this contrary LaTex.