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)
(I'm assuming A is a subset of the reals.)
Claim: A is a subset of (-inf, b*]
Proof: b* = sup A.
Claim: (-inf, b*) is a subset of A
Proof: Because b* = sup A, there exists . Because A not bounded below, there exists s.t. .
Let . Then there exists s.t. and s.t. .
Since , it follows by A convex that x in A. Thus .
Claim: If b* in A, then . If b* not in A, then .
Proof: From previous two claims. Know that .
If b* in A, then , and b* in A, so that their union, , is a subset of A.
But then , thus .
If b* not in A, then and , so A is contained in their intersection, .
Thus , thus .