# real analysis

• Aug 17th 2009, 12:37 PM
flower3
real analysis
Give an example to show that it is not true ,ingeneral ,that :
sup(AB)=sup(A)sup(B) and inf (AB)=inf(A)inf(B)
• Aug 17th 2009, 01:23 PM
Bruno J.
Hint for the first one:

$\displaystyle A: \: 1,-1,1,-1,...$
$\displaystyle B: \: -1,1,-1,1,...$
• Aug 17th 2009, 02:17 PM
putnam120
Also consider $\displaystyle a_n=\cos\left(\frac{n\pi}{2}\right)$ and $\displaystyle b_n=\sin\left(\frac{n\pi}{2}\right)$.
• Aug 17th 2009, 02:57 PM
Plato
Quote:

Originally Posted by Bruno J.
hint
$\displaystyle A: \: 1,-1,1,-1,...$
$\displaystyle B: \: -1,1,-1,1,...$

But that depends upon how $\displaystyle AB$ is defined.
If it is defined by $\displaystyle AB=\left\{ {xy:x \in A\;\& \,y \in B} \right\}$ then $\displaystyle \sup(A)=\sup(AB)=\sup(B)$

How is $\displaystyle AB$ defined?
• Aug 18th 2009, 12:32 AM
Opalg
Quote:

Originally Posted by flower3
Give an example to show that it is not true ,ingeneral ,that :
sup(AB)=sup(A)sup(B) and inf (AB)=inf(A)inf(B)

How about $\displaystyle A = \{1,2\},\ B=\{-1,-2\}$ ?