Question 2. In general, we are given continuous best response functions f (ac) and g(y). Furthermore, suppose that there is an interval [(1, ] such that a s M) s 5 and a : g s 6 for an 27,1; 6 [06,5] i.e. the ranges of f and g on [a, ] are still contained in [04, ,3]. We will prove that in this case there is always a Nash equilibrium (3%, g) satisfying a S 5:, g] S B. (a) First prove the following statement. A xed point of a function h(m) is a number c in its domain such that h(c) = c. Use the Intermediate Value Theorem to prove that any continuous function h(x) with domain [04, ] and range in [a, B] must have a xed point. (Hint. Consider the function H (:13) : h(:z:) x. Show that H (a) 2 0, H () 3 0 and there must be a number 0 E [a, B] such that H(c) = 0.) (b) Consider the function h(x) = g(f(x)). By (a), show that h(x) has a fixed point xo E [a, B], i.e. g(f(x0)) = x0. Then explain that (x, y) = (x0, f(x0) ) is a Nash equilibrium