your work is incomplete. showing it is a lower bound is not enough. you need to show it is the greatest lower bound. there are ways to do that elegantly. i will use the fact that for .

Let and be as defined. By the definition of infimum, we have that for all . so that for all . but that means is an upper bound for . thus, since the supremum is the least upper bound, we must have ........(1)

Also, for all by the definition of supremum. but that means for all , so that is a lower bound for the set . since the infimum is the greatest lower bound, we must have that ...........(2)

By (1) and (2) we have , as desired

yes, we can assume the supremum and infimum exist. they are and in the extreme casesNow that I look back on my work so far, I think I am wrong in assuming right away that a supremum exists. Do I have to first prove the existence of a supremum than show it is equal to the infimum? I am really stuck here.

Any help is greatly appreciated. Thank you for your time!