I solving a proof relating sup(AB) and Binf(A)

[cpl1992]

Homework Statement

Let A be a bounded nonempty subset of the set of all real numbers (R). B exists in R and B<0. Let BA= {Ba: a exists in A} Prove sup(BA)=Binf(A)

Homework Equations

We are able to use the ordered field axioms, Archemedian Property ect..

The Attempt at a Solution

I know that I need to show
that sup(BA)<=Binf(A) and sup(BA)>=BinfA
and If A is bounded then y<b where b is a bound for all y that exist in A

 

[happysauce]

Homework Statement

Let A be a bounded nonempty subset of the set of all real numbers (R). B exists in R and B<0. Let BA= {Ba: a exists in A} Prove sup(BA)=Binf(A)

Homework Equations

We are able to use the ordered field axioms, Archemedian Property ect..

The Attempt at a Solution

I know that I need to show
that sup(BA)<=Binf(A) and sup(BA)>=BinfA
and If A is bounded then y<b where b is a bound for all y that exist in A

What is the definition of supremum and infimum?

 

[cpl1992]

Supremum is the lowest upper bound of the set and infimum is the highest lower bound of the set

 

[happysauce]

Supremum is the lowest upper bound of the set and infimum is the highest lower bound of the set

Ok, try thinking in terms of elements, that is, suppose i call β the supremum, how does that relate to elements in the set? And by supremum I'm referring to the supremum of set A.
Think about it, we have the ordered field axioms, a negative number, and the definition of sup and inf.

 

[cpl1992]

I still seem to be confused as to how B would relate to the elements in the set. If B is sup(A) then this is saying it is the lowest upper bound of A. If this is the lowest upper bound then B could be either less than or greater than the set of BA itself correct?

 

[happysauce]

I still seem to be confused as to how B would relate to the elements in the set. If B is sup(A) then this is saying it is the lowest upper bound of A. If this is the lowest upper bound then B could be either less than or greater than the set of BA itself correct?

Try thinking of an example. Suppose I have the set A=(1,2) and B=-1
What is the supremum of B*A? What is B * infimum of A?

Does this help?

Also beta is not B, sorry i should have used a better letter. Beta is the supremum of the set.