# Thread: Supremums: sup AC = supA supC

1. ## 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!

2. 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 $\displaystyle a\in A$ and $\displaystyle c\in C$ then $\displaystyle a\leqslant \sup\text{ }A$ and $\displaystyle c\leqslant \sup\text{ }C$ and so $\displaystyle ac\leqslant\sup\text{ }A\cdot\sup\text{ }C$ it follows that $\displaystyle \sup\text{ }AC\leqslant\sup\text{ }A\cdot\sup\text{ }C$.

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