LetDbe the set of rational numbers that may be expressed ask/2n, wherekZandnN. The goal of this problem is to show thatDis dense inRi.e.D=R.LetDn={k/2n|k= 0,1,2,...,2n}, wherenN, and setD=n=1Dn. 1.Prove thatD[0,1]. 2.Letx[0,1]. Split the interval [0,1] into two equal subintervals and callI1= [a1,b1] the part containingx. Repeat the split withI1: divideI1into two equal parts and nameI2= [a2,b2] the portion containingx. Afterksplits, we havexIk= [ak,bk]. 3.Prove by induction that for allkN,ak, bkDandbkak= 2k. 4.Prove that for allkN,|xak|2k. 5.Prove thatakxask. 6.Prove that [0,1]D. 7.Prove thatD= [0,1]. 8.LetyR.Prove that 0ybyc<1, wherebcis the integer part or floor function. 9.Prove that there is a sequence{uk}ofDsuch thatukyask. 10.Prove thatD=R.