1. The problem statement, all variables and given/known data I'm reading Goldrei's Classic Set Theory, and I'm kind of stuck in the completeness property proof, here is the page from googlebooks: https://books.google.com/books?id=1dLn0knvZSsC&pg=PA14&lpg=PA14&dq="first+of+all,+as+a+is+non+empty"&source=bl&ots=6fwW7jd8i4&sig=txERO1ZYpOKkJ_SVdY82LNMs3io&hl=en&sa=X&ved=0ahUKEwjmxLn_wIbKAhVX-mMKHdPaCSkQ6AEIHzAA#v=onepage&q="first of all, as a is non empty"&f=false 2. Relevant equations - 3. The attempt at a solution Ok, I guess I understand the first part regarding α=UA. What I don't get is why are we trying to prove that UA≠ℚ? Also, what does he mean when he says: "As s∈ℝ there is some rational q such that q∉s", how is that possible? if s is a real number, it should include all rational numbers, right? Unless what's meant by that is that there is a rational q that's greater than s, and thus doesn't belong to s?