# Thread: supS = -inf(-S) But how?

1. ## supS = -inf(-S) But how?

Hi, our instructor left this question as an exercise for us. And I first need to prove
supS ≥ -inf(-S)
and
-inf (-S) ≥ supS

I think I could do st on the first part.

Say x S

supS ≥ x

For -x -S

-x ≥ inf(-S) If we multiply it with - ;

-inf(-S) ≥ x
Then we get
supS ≥ x
-inf(-S) ≥ x

Since the greatest value of x is supS, then supS ≥-inf(-S).

But I cannot find a way to prove the reverse. Any thoughts? Thanks in advance.

2. yes do the same for the set (-S) and you get the inequality that sup(S) $\le$ -inf(-S); combine two inequalities and you have your equality.