Querrian ayudarme con lo siguiente:
A,B, subconjuntos de R acotados y no vacíos
a)Si SupA<0 y SupB<0
Demostrar:
Inf(AB)=SupA.SupB
gracias
You probably want to show:
$\displaystyle \inf AB \leq \sup A * \sup B$ and,
$\displaystyle \inf AB \geq \sup A * \sup B$.
Hint: (contradicción).
Hmm.. Maybe I can get you started, one of the directions seems simple.
$\displaystyle a \leq \sup A,$ $\displaystyle \forall$ $\displaystyle a \in A$.
$\displaystyle b \leq \sup B,$ $\displaystyle \forall$ $\displaystyle b \in B$.
Now,
$\displaystyle a*b \leq \sup A * \sup B$. [MISTAKE?]
Also,
$\displaystyle \inf AB \leq a*b,$ $\displaystyle \forall$ $\displaystyle a \in A,$ and $\displaystyle b \in B$.
So,
$\displaystyle \inf AB \leq a*b \leq \sup A * \sup B,$ and thus
$\displaystyle \inf AB \leq \sup A * \sup B$
Hmm, I dunno bout this:
-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
show that:
infAB = supA.supB
Here's what I would do:
Since supA< 0 it means there is a negative number w such that $\displaystyle a\leq w<0$ for all a in A. Also, there is a v<0 such that $\displaystyle b\leq v<0$ for all b in B.
So $\displaystyle ab\geq vw$ 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.