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.
The Attempt at a Solution
I have proven a)
Let b ∈ B. Let u ∈ U. Then by the definition of upper bound, (b,u) ∈ R.
For the proof of b), you obviously assume the antecedent of the statement to be shown. I can't, however, seem to make out the logical form of this. Also, it appears you must use part a) in the proof of part b) as well. I have an intuitive understanding of the idea of the statement to be proven, I am just having a hard time (in-)formalizing it.
The logical form of the goal- i would think- is (∀b ∈ B)((b,x) ∈ R) and (∀u ∈ U)((x,u) ∈ R); I assume this means that x is an upper bound of B AND that x is the least upper bound of B (hence, x is the least upper bound of B).
So naturally, my proof thus far is:
Suppose x is the greatest lower bound of U. Let b ∈ B....