Least Upper Bound proof.... 1. The problem statement, all variables and given/known data Suppose A is a nonempty set that has x as an upper bound. Prove that x is the least upper bound of the set A iff for any E>0 there exists a y in A such that y>x-E 2. Relevant equations None 3. The attempt at a solution The forward where you assume that x is the least upper bound is very easy, but I'm having some trouble proving the reverse..... This is what I have so far.... Let x be an upper bound of A, and choose a point z in A. If x is an upper bound of A, then x+z is also an upper bound.