Will you please just state exactly what you are required to prove?
Is it, “If a set has a lower bound, then it has an infimum”?
If so, do you know that if a set has an upper bound them it has a supremum?
If I use to show E is nonempty and bounded above, then E has a supremum. What would I use to show: If E is nonempty and bounded below, then E has an infimum. Would work? When I tried this the induction part doesn't make sense to me. For example by induction, then, there exists integers least in such that and to me that doesn't seem to make sense for something bounded below.
How did you "use " to prove that non-empty set, with upper bound, has a supremum? By using monotone convergence? You can do something similar by multiplying by -1 to make the increasing sequence into the decreasing sequence . Or do as Plato suggested and use the fact about supremum to prove infimum- again multiply by -1 to reverse the order.