Completeness axiom/theorem and supremum

  • Thread starter Treadstone 71
  • Start date
  • Tags
    Supremum
In summary, A and B are bounded nonempty sets of real numbers and C={ab:a in A, b in B}. By the completeness axiom/theorem, A and B have suprema. For all e>0, there exists a in A and b in B such that z-e<a and y-e<b. Multiplying the two inequalities, we have zy-ze-ye+e^2 < ab. To show that zy is an upper bound on C, we can use the definition of sup as the maximum of the accumulation points. Given d>0, we can find e>0 such that 0<ey+ez-e^2<d. Since z and y are the least upper bounds on A and
  • #1
Treadstone 71
275
0
"Let A and B be bounded nontempty sets of real numbers. Let C={ab:a in A, b in B}. Prove that sup(C)=sup(A)sup(B)."

Here's what I've done so far:

By the completeness axiom/theorem A and B have suprema. Let sup(A)=z and sup(B)=y. For all e>0, there exists a in A and b in B such that z-e<a and y-e<b. Multiplying the two inequalities and we have

zy-ze-ye+e^2 < ab

I'm stuck here.
 
Physics news on Phys.org
  • #2
Treadstone 71 said:
"Let A and B be bounded nontempty sets of real numbers. Let C={ab:a in A, b in B}. Prove that sup(C)=sup(A)sup(B)."
Here's what I've done so far:
By the completeness axiom/theorem A and B have suprema. Let sup(A)=z and sup(B)=y. For all e>0, there exists a in A and b in B such that z-e<a and y-e<b. Multiplying the two inequalities and we have
zy-ze-ye+e^2 < ab
I'm stuck here.

It would be a good idea to show first that zy is an upper bound on C!
You are using the fact that since z is the least upper bound on A there must be an a between z-e and z and since y is the least upper bound on B there must be a b between z-e and z. What does that tell you about there being an ab between zy- e and z?
 
  • #3
step 1 show it's an upper bound,

step 2 show it is a least upper bound which can be messy.

it's easier to use the definition of sup as the maximum of the accumulation points.

You need to show that give d>0 you can find e greater than zero such that 0<ey+ez-e^2<d, so do that.
 
  • #4
I don't know, but I want to arrive at the conclusion that zy-x<ab for all x>0. If I could prove that zy-(?)<ab where (?) is positive, then I'm done, since I can let x=(?).
 
  • #5
we may suppose e<z and e<y, can you see how that might help?
 
  • #6
e>0. z and y could be negative.
 
  • #7
How can the supremum of a set of positive numbers be negative?
 
  • #8
A and B aren't sets of strictly positive numbers.
 
  • #9
Then the proposition is trivially false.
 
  • #10
You're right. This is odd considering it's an archived analysis final exam.
 
  • #11
The odd thing is that at one point i even checked back to make sure that these were sets of positive numbers just so i didn't make a mistake and i am convinced that i remember reading that you wrote they were positive real numbers. Odd.
 

1. What is the completeness axiom/theorem?

The completeness axiom, also known as the completeness theorem, is a fundamental concept in mathematics that states that any nonempty set of real numbers that is bounded above must have a least upper bound, or supremum. This means that there is a single number that is greater than or equal to all the numbers in the set, but is still the smallest such number.

2. Why is the completeness axiom/theorem important?

The completeness axiom is important because it guarantees the existence of a solution to many mathematical problems, particularly in calculus and analysis. It also allows for the creation of a rigorous and consistent mathematical system, which is essential for the development of advanced mathematical theories and applications.

3. What is the difference between the completeness axiom and the supremum?

The completeness axiom is a fundamental principle in mathematics, while the supremum is a specific number that satisfies the completeness axiom. In other words, the completeness axiom is a general concept, while the supremum is a specific instance of that concept.

4. How is the completeness axiom/theorem used in real-life applications?

The completeness axiom/theorem is used in many real-life applications, particularly in the fields of physics, engineering, and economics. For example, it is used in optimization problems to find the most efficient solution, and in modeling systems that involve continuous quantities, such as fluid dynamics.

5. Are there any limitations to the completeness axiom/theorem?

Yes, there are some limitations to the completeness axiom/theorem. One limitation is that it only applies to real numbers and cannot be extended to other number systems, such as complex numbers. Another limitation is that it does not provide a way to explicitly find the supremum, but only guarantees its existence.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
7K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
466
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
790
  • Calculus and Beyond Homework Help
Replies
14
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
2
Replies
38
Views
8K
Back
Top