10. (Brouwer Fired-Point Theorem) A point c is called a fixed point of a function f(r) iff f(c) = c. Let I = [0, 1], and let f(x) be a continuous function mapping I into I; that is, 0 < f(x) <1 for all a with 0≤x≤ 1. Prove that f(x) has a fixed point in I. (Hint: Draw a picture. Let g(x) = x - f(x). Use the Intermediate Value Theorem.)