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

  • Thread starter Thread starter cpl1992
  • Start date Start date
  • Tags Tags
    Proof
Click For Summary

Homework Help Overview

The problem involves proving a relationship between the supremum of a transformed set BA, where B is a negative constant and A is a bounded nonempty subset of real numbers, and the infimum of the original set A. The context is rooted in real analysis, specifically dealing with concepts of supremum and infimum.

Discussion Character

  • Conceptual clarification, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the definitions of supremum and infimum, with some attempting to relate these definitions to the elements of the sets involved. Questions arise regarding how the negative constant B interacts with the bounds of the sets.

Discussion Status

The discussion is ongoing, with participants exploring the definitions and relationships between the sets. Some guidance has been offered regarding thinking in terms of elements and examples, but there is no explicit consensus on the approach to the proof.

Contextual Notes

There is a noted confusion regarding the relationship between B and the elements of the set BA, as well as the implications of B being a negative number. Participants are encouraged to consider specific examples to clarify their understanding.

cpl1992
Messages
4
Reaction score
0

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
 
Physics news on Phys.org
cpl1992 said:

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?
 
Supremum is the lowest upper bound of the set and infimum is the highest lower bound of the set
 
cpl1992 said:
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.
 
Last edited:
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?
 
cpl1992 said:
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.
 

Similar threads

Hermitian Matrix and Commutation relations
  • · Replies 2 ·
Replies
2
Views
3K
Prove B is invertible if AB = I
  • · Replies 40 ·
2
Replies
40
Views
6K
Proving the Evenness of Elements Not Equal to Their Own Inverse in Finite Groups
  • · Replies 11 ·
Replies
11
Views
3K
If A is in B then sup(A) < sup(B)
  • · Replies 1 ·
Replies
1
Views
2K
For a set S, there is always a sequence converging to sup(S)
  • · Replies 14 ·
Replies
14
Views
3K
Prove that sup(ST) = sup(S)sup(T)
  • · Replies 7 ·
Replies
7
Views
3K
Set of least upper bounds multiplied by a constant
  • · Replies 1 ·
Replies
1
Views
1K
Proof of Subgroup Property for Cyclic Group G: Homework Help
  • · Replies 3 ·
Replies
3
Views
2K
Supremum, Infimum (Is my proof correct?)
  • · Replies 3 ·
Replies
3
Views
2K
If a.a=a prove R is commutative
  • · Replies 4 ·
Replies
4
Views
2K