Suppose that then if follows that .
That mean that is an upper bound of the set .
Say that then .
By the definition of infimum
That means that .
This means that no number less that is an upper bound of .
Therefore .
Prove, if S is a nonempty subset of real numbers that is bounded below then
inf(S)= -sup{-s: s in S}.
I have shown that the {-s:s in S} is bounded below and I let u = sup of that set but I do not know how to show -u = inf(S).