1. The problem statement, all variables and given/known data Show that: Let S be a subset of the real numbers such that S is bounded above and below and if some x and y are in S with x not equal to y, then all numbers between x and y are in S. then there exist unique numbers a and b in R with a<b such that S is one of the intervals (a,b), [a,b), (a,b], or [a,b]. 2. Relevant equations 3. The attempt at a solution Assume if x and y are elements of S with x not equal to y, then all numbers between x and y are in S and S is bounded above and below. Thus there exists a M, N such that M is greater than or equal to the maximal element of S and N is smaller than the minimal element of S. Also all elements between x and y are inside (M,N).