Exercise 2. One way to define an order on R* is as follows. Given x, y Rk, let x < y if Ti < y; at the first coordinate i in which they differ. That is, x < y if x1 < y1, or * = y and x < y2, or * = y and = y and and k-1 = yk-1 and k < Yk- This is called dictionary order or lexicographic order; essentially, we compare x and y as if they were "words" of length k, where each "letter" is a real number. (a) Prove that this is an order on Rk. (b) If k 2, does Rk with this order have the least-upper-bound property? Give a proof if true or a counterexample if false.