You probably want to show:
Hmm.. Maybe I can get you started, one of the directions seems simple.
-5<-3 and -2<0 but (-5)*(-2)>(-3)*0
I don't speak the language of the original post, but I think it translates as this in English:
If A and B are non-empty subset of real numbers such that
supA< 0 and supB< 0
infAB = supA.supB
Here's what I would do:
Since supA< 0 it means there is a negative number w such that for all a in A. Also, there is a v<0 such that for all b in B.
So for all a in A and b in B, which we can do because of the signs of everything. This shows that vw is a lower bound for AB, i.e. supA.supB is a lower bound for AB. All that remains to show is that vw is the largest lower bound. Try to see if you can show this.