Let D be the set of rational numbers that may be expressed as k/2", where k Z and n N. The goal of this problem is to show that D is dense in Ri.e. D= R. Let D = {k/2" | k= 0,1,2,...,2") and set D = =DR. 1. Let A and B be subsets of R. Show that A C B = A SB. 2. Prove that for all n N, D, C [0, 1]. 3. Prove that D[0, 1]. 4. Prove that D[0, 1]. 5. Let x [0, 1]. Split the interval [0, 1] into two equal subintervals and call I = [a, b] the part containing z. Repeat the split with I: divide I; into two equal parts and name 12 = [02, 6] the portion containing z. After k splits, we have r [a, b]. a) Prove by induction that for all k N. a, b D and bak = 2-k b) Prove that for all k N, 0z-a 2-*. c) Prove that a ras k oo, and b I ask o. 6. Prove that [0, 1] D... 7. Prove that D = [0, 1]. 8. Let y R. a) Prove that 0 y - [y]