1. The problem statement, all variables and given/known data Prove if an ordered set A has the least upper bound property, then it has the greatest lower bound property. 2. Relevant equations Definition of the least upper bound property and greatest lower bound property, set theory. 3. The attempt at a solution Ok, I think that my main problem with this proof is knowing exactly what I'm allowed to do while constructing proofs. I have to prove that the least upper bound property implies the greatest lower bound property, and to start out with this I tried to write out some definitions and see what I can come up with and see if that gives me any insight to how one implies the other, I know that A has the least upper bound property if and only if it has the greatest lower bound property, that's given by the book. So to start constructing this I tried something like this: Let A0 ⊂ A, and A0≠∅, so here I'm defining a restriction of the set A, calling it A0 and saying that it is not empty. Then I tried this: Let S be the set of all upper bounds b, b|b∈A and b≥x for every x∈A0, so here I'm trying to make the set of all upper bounds, call it the set S, and say that every element of the set S is greater than or equal to x∈A0, but this is as far as I can get, after this I'm lost. Like I said, I'm not even sure if I'm "allowed" to do what I did. Is there something I'm missing from this? I think that next I would try to show that there is a inf A0, but I don't know how to get that out of the construction that I've made.