Let f : (a,b) –R. Prove that if f is differentiable at xo € (a, b), then the sequence {n(f(xo + ±) – f(xo))}-1 converges to f'(xo). n=1

7. Let fn: R → R be a sequence of continuous functions which converges uniformly to a function f : R → R. Let (n) be a sequence of real numbers which converges to x E R. Show that fn(xn) → f(x).

2. Let f RR be a function with the property that there exists K € (0, 1) such that, for all x, y € R, f(x) − f (y)| ≤ K|x - y. Such a function is called a contraction of contraction factor K. (a) Prove that f is continuous. (b) Let c ER and (xn) be the sequence defined by x1 = C, Xn+1 = 0 f(xn) for all n ≥ 1. n+1- xn| ≤ Kn-¹|x2 - x₁| for all n ≥ 1. Kn-1 i. Prove, by induction, that ii. Prove that, for all m n ≥ 1, xm- xn| ≤ 1- K iii. Deduce that (n) is Cauchy, and hence converges. iv. Hence prove that f has a fixed point. v. Prove that the fixed point of f is unique. (You have just proved a special case of the Contraction Mapping Theorem.) (c) Write down a continuous function g: R → R which has no fixed points. Verify that it is not a contraction. = |x₂ - x11.

4. Let (#1, 1) = (0,0) and let (an+1, Yn+1) = (2 + Yn, 3 – Tn) for n > 1. Prove that (&n, Yn) is a contractive sequence and find its limit.