Why is it stated in the solution that $SupB\in{B}$? By the completeness axiom the set B has a supremum, but that does not imply that it's the maximal element of the set. My intuition is telling me whats written is correct, but I don't know how to state it.
It is given that A has a lower bound, so by the completeness axiom A has an inf. A's inf is defined as its maximal lower bound, which is exactly saying that if A has an inf , the set of its lower bounds has a maximal element, so $supB=maxB\in B$.