If A and C are subsets of R, let AC={ac: a E A, c E C}. If A and C are bounded and the sets consist of strictly positive elements, prove that sup AC = supA supC.

Attempt:

a≤(sup A) and c≤(sup C)

=> ac ≤ supA supC for all (ac) E (AC)

Hence supA supC is an upper bound for the set AC.

But how to prove that sup AC = supA supC?

Any help is appreciated!