I was playing with my calculator, had an arbitrary number on the screen, then pressed 1/x ,then sqrt, then 1/x, and so forth. I noticed it converged to 1.0 after a few back-and-forths. So, it occurred to me that I did not know how to prove that such procedure converges to one for an arbitrary number different from zero.
And when playing with different function pairs such as, $x^2$ and $\sqrt{x}$ (this time using a computer), I notice that I get other bounded patterns.
So, my question is regarding the convergence properties of a function of the following kind:
$$ y_{k+1}=f_{k\mod{2}}(y_k) $$
where both $f_0$ and $f_1$ are given functions, $y_0$ is a given real number, and $k=0, 1,\ldots,\infty$.
What can be said about $y_k$ as $k\to\infty$?
Pardon my lack of knowledge! I am sure someone has looked at this!
P.S. Some visuals for those so inclined.
note: migrated from mathoveflow at the request of @Anthony Quas
