Thread: Supremums: sup AC = supA supC

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!

Originally Posted by kingwinner

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!

Let and then and and so it follows that .

That finishes up the rest of your initial thought, care to try the second part?