1. The problem statement, all variables and given/known data Okay, this is essentially the same question I had in an earlier thread, but i am trying to make my questions and uncertainties more clear for more accurate assistance: Suppose R is a partial order on A and B ⊆ A. Let U be the set of all upper bounds for B. a) Prove that every element of B is a lower bound for U. b) Prove that if x is the greatest lower bound of U, then x is the least upper bound of B. 2. Relevant equations My trouble is in proving part b). Also, I feel part a)- which i have proven below- is used in the part b) proof, I just can't seem to piece the two together appropriately. 3. The attempt at a solution Here is my proof for part a) Let b ∈ B. Let u ∈ U. Then by the definition of upper bound, (b,u) ∈ R. Since u was arbitrary, b is a lower bound of U. Since b was arbitrary, every element of B is a lower bound of U. To prove part b) obviously you assume the antecedent of the statement to be proven. That is, x is the greatest lower bound of U. You can also assume the result obtained in part a). Now our goal is x is the least upper bound of B. Logically this translates to for all b in B, (b,x) is an element of R. Also, for all u in U, (x,u) is an element of R. So I am trying to prove these separately. My confusion begins here: Does the antecedent mean x is a lower bound of U AND x is the greatest element of the set of lower bounds of U (which, as far as i can come up with, is A\U)? How do you show x is an upper bound of B? I feel this is a start to the rest of the proof.