Q:

A fixed-point theorem by Zamfirescu

I am having a trouble with understanding the proof of a fixed-point theorem by Zamfirescu.
Could somebody please explain how the inequality in the inner, pink rectangle is obtained from the previous one? I see no reason for this to be true.

A:

Isn't this just arithmetic? Expand the RHS and collect terms, giving $(1-\beta)d(f^{n+1}(x_0),f^{n+2}(x_0))$ on the LHS and $\beta d(f^n(x_0),f^{n+1}(x_0))$ on the right. Since $\beta < \frac{1}{2}$, $1-\beta>0$ and the result follows.

