Fix some a > 0. Consider the function G : (0,) (0,) given by a G(y) == y + (a) Show that G is continuously differentiable, and that G' (y) < 0 for y < a and G' (y) > 0 for y> a. Use this to show that G achieves an absolute minimum at y = a. (Remark: Feel free to use the fact that y' = 1 and (1/y)' = 1/y) (b) Let I = [a,). Show that the restriction G|, is Lipschitz continuous, with some Lipschitz constant L < 1. (Hint: For an easy way to find the Lipschitz constant, look at HW7) (c) Define yo= max{a, 1}. Note that yo a. Define iterates Yn = G(Yn-1) Show that the sequence of iterates {yn} satisfies yn for all n N. Then, show that for all k > n, the following chain of inequalities holds (i.e. explain where each line comes from): | Yk - Yn L" | Yk-n - Yo k-n