Let an be the sequence defined by 112113231141234115Prove th

Let {an} be the sequence defined by {1,1/2,1,1/3,2/3,1,1/4,1/2,3/4,1,1/5,.....}.Prove that each rational number r in the interval (0,1],there are infinitely many indices such that an=r

Solution

Given: an = {1,1/2,1,1/3,2/3,1,1/4,1/2,3/4,1,1/5,.....}.

      here have either 1 or rational number

        these rational numbers are:1/2=0.50

                                              1/3=0.33

                                              2/3=0.66

                                              1/4=0.25

                                               1/2=0.50

                                               3/4=0.75

                                               1/5=0.20

          all these are below 1 or (0 to1) but we have 1 in the an = {1,1/2,1,1/3,2/3,1,1/4,1/2,3/4,1,1/5,.....}.

         so that each rational number r in the interval (0,1],there are infinitely many indices such that an=r