real analysis

• Aug 17th 2009, 01:37 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, 02:23 PM
Bruno J.
Hint for the first one:

$A: \: 1,-1,1,-1,...$
$B: \: -1,1,-1,1,...$
• Aug 17th 2009, 03:17 PM
putnam120
Also consider $a_n=\cos\left(\frac{n\pi}{2}\right)$ and $b_n=\sin\left(\frac{n\pi}{2}\right)$.
• Aug 17th 2009, 03:57 PM
Plato
Quote:

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

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

How is $AB$ defined?
• Aug 18th 2009, 01:32 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 $A = \{1,2\},\ B=\{-1,-2\}$ ?