Can Two Sets' Supremum Equal the Product of Their Individual Supremums?

[emob2p]

Hi,

I'm trying to prove that b^{r+s}=b^r*b^s for any real r,s where b^r = sup{b^t:t \leq r} and t is rational. (This is prob 1.6f in Rudin)

My question. Can one show that for two sets X and Y:

sup(XY)=(supX)(supY) where XY = {x*y: x\in X, y\in Y}

Thanks,
E

 

[morphism]

Try playing with positive and negative numbers.

 

[emob2p]

=> Counterexample very easy