The least upper bound property says that, "Every nonempty subset of $A$ that $is$ bounded above has a least upper bound." The great lower bound property is defined similarly, and it's not difficult to show that one property holds if and only if the other holds. The key lies in the assumption that we only deal with subsets that are bounded below to show they must have an inf.
On the other hand, an old homework (question 5) of mine says to prove that if every subset of a poset has an inf then every subset must also have a sup; I just don't believe this because one must first prove that such subsets are even bounded above (and this was not stated in the problem).
There's also this post on Stack Exchange which has no answer, though the OP believes to have found an answer, it doesn't look good.
Is it fair to say that the homework question is just wrong and what was intended to be said is something like, "Prove the equivalence of the lub and glb properties."?