2. [This question is essentially question 2.22 in the textbook.] Let A, G, and H be the arithmetic, geometric, and harmonic means of two positive real numbers a and b: -1 2ab A(a, b) = a+b G(a, b) = Vab, H(a, b) = 2 2a 2b a +b (a) Prove that A(a, b) - H(a, b) = (b - a)2/2(a + b). (b) Define two sequences recursively as follows. Let a1 = H(a, b), b1 = A(a, b). For each n, define an+1 = H(an, bn), bn+1 = A(an, bn) Prove that {[an, bn]} is a sequence of nested closed intervals, and deduce that there is some c E n, [an, bn]. (c) Prove that c = G(a, b). [Hint: Show anby, = G(a, b)2 for all n.]