The exercise:

Suppose that R is a partial order on A, B1 ⊆ A, B2 ⊆ A, x1 is the least upper bound of B1, and x2 is the least upper bound of B2. Prove that if B1 ⊆ B2, then (x1,x2) ∈ R.

(looking for an informal proof here)

My attempt at solving:

I have been finding this problem somewhat difficult. After assuming B1 ⊆ B2, I have been trying to construct a proof by cases- however, I am not even certain that this is the best way to proceed. The cases I have attempted are x1 ∈ B2 or x1 ∉ B2, but i haven't found a way to complete the proof this way, and so i am beginning to doubt this proof strategy (by cases). As such, I have also attempted a proof by contradiction, but this doesn't seem to make it too far either.

My main questions are:

1) If this IS a proof that should proceed via cases, are there any suggestions as to which cases should be used?

2) If this is not a proof by cases, what method of proof should be used, and in what ways does the problem suggest this?

I have been practicing proofs in set theory for about a year and am now reviewing for a credit by examination test I will be taking this semester by performing some of the problems which occur later in the problem sets in Velleman's text on proofs/set thoery. If you have any insight, I would also like to ask your opinion as to the degree of difficulty you believe this problem is (I'm interested in assessing my mastery of the subject). The proofs which appear in the examples within the text and earlier within problem sets I can easily achieve, but the later ones usually leave me thinking for many hours, and sometimes even a day or two (oftentimes without much progress toward the solution).

Any help is appreciated