I don't understand your question. What you've given is the definition of the infimum of a set of real numbers. From what I could follow of your answer, I think you are trying to prove that given every bounded above set of real numbers has a supremum, every bounded below set has a infimum as defined above which is unique. Then yes, your answer seems correct but it is not very clear.

Everything revolves around applying negatives to our established inequalities from the supremum as you were considering, but it seems like you were going about it kind of backwards. Half the battle in proof writing is understanding the answer, and the other half is making your answer understandable to others. Here is how it should start:

Let

be a nonempty set of real numbers which is bounded below. Then there exists a real number

such that

for all

. Then

for all

. Then for

,

for all

. Then

is bounded above and obviously nonempty, so there exists a supremum...