If sup B > sup A, then prove that there is an element b of B that is an upper bound for A.

My work: I split this into two cases, that where B has a maximum, and that where it doesn't. If it has a maximum, then we can set b = sup B = max B, and clearly this is an upper bound for A since sup B > sup A. But I'm lost for the second case...

In general, I'm having trouble with these supremum proofs when it comes to the case where the set doesn't have a maximum. Any help?