Let A be a nonempty subset of R (the reals) and let -A = {-x : x in A}.

Prove that inf A = -sup (-A).

My work:

Proof: Let A be non empty subset of R and -A = {-x : x in A}. If -x is in -A then

-x < sup (-A) by the definition of supremum. This implies x > -sup (-A), and so

-sup (-A) is a lower bound of A.

Now 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!