What exactly are you having difficulties with? This is a pretty straightforward result. How about I get you started and you finish?
Let be nonempty set bounded below, by the completeness axiom we know that exists. So we can see that for any , and if the is not a lower bound of ...now multiply by negative one, noting that multiplying a set by negative one is equivalent to multiplying each element by negative one. What does this do to ? What can we conclude from this?
I hope that helps...any problems just ask
P.S. Frome one analysis novice to another I have found that problems like this can look like dragons but almost always end up being puppies...exhaust every idea...leave no stone unturned...some other trite expression...just get your hands dirty.