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.