### Functions and Sequences of Iterates

Let f be a function defined on [a,b] and suppose that z is a point in (a,b) such that f(z) = z. Further suppose that there is a number alpha < 1 such that f ' (x) < alpha for all x contained in (a,b) and that 0 < f ' (x) for all x contained in (a, b). a.) Prove that if z<x, then z<f(x) and that f(x) - z < alpha(x-z).