A is a convex, nonempty set. A is not bounded below and is bounded above. B* = sup A.

Prove:

Claim 1: (-inf, b*) is a subset of A

Prove:

Claim 2: A is a subset of (-inf, b*]

Case 1: b* is not an element of A

Prove:

Claim 2.1: A = (-inf, b*)

Case 2: b* is an element of A

Prove:

Claim 2.2: A = (-inf, b*)

(There are 4 proofs here)