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__Fact:__If A and C are subsets of R, let AC={ac: a E A, c E C}. If A and C are bounded and A and C consist of strictly positive elements only, then sup AC = supA supC.I am trying to understand and prove this fact, and this is what I've got so far...

A, C bounded => supA, supC exist (by least upper bound axiom)

0<a≤supA for all a E A

0<c≤supC for all c E C

=> 0<ac≤supA supC for all ac E AC

=> supA supC is an upper bound of AC

=> supAC≤supA supC

But how can we prove the other direction? I think we somehow have to use the fact "for all ε>0, there exists a E A such that sup A - ε < a."

At the end, we have to show that for all ε>0, supA supC - ε is NOT an upper bound for AC. But I'm not sure how it is going to work out here in our case.

Does anyone have any idea?

Thanks for any help!