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