Proving Unboundedness of B if A is Unbounded

  • Thread starter Seacow1988
  • Start date
In summary, to prove unboundedness of B if A is unbounded, you need to show that for any positive number M, there exists a number x in B such that x is greater than M. The relationship between A and B in proving unboundedness is that A must be unbounded in order for B to be unbounded. For example, if A is the set of all positive integers and B is the set of all even integers, since A is unbounded, B is also unbounded. It is not possible for B to be unbounded if A is bounded, as B is a subset of A. Proving unboundedness of B if A is unbounded can have implications in mathematical proofs and real-world applications in
  • #1
Seacow1988
9
0
If A, B are nonempty subsets of R and A is a subset of B, how can you prove that: if A is unbounded, B is unbounded?
 
Physics news on Phys.org
  • #2
Suppose to the contrary that B is bounded. So there exists a real number M such that |b| < M for all b in B. Do you see where the problem lies?
 
Last edited:
  • #3
If you understand the definition of 'boundedness' and 'subset' then this is trivial.
 

1. How do you prove unboundedness of B if A is unbounded?

To prove unboundedness of B if A is unbounded, you need to show that for any positive number M, there exists a number x in B such that x is greater than M. This means that B has no upper bound and is therefore unbounded.

2. What is the relationship between A and B in proving unboundedness?

In order to prove unboundedness of B, you must first know that A is unbounded. This is because if A is bounded, then B cannot be unbounded.

3. Can you provide an example to illustrate this concept?

Sure, let's say that A is the set of all positive integers and B is the set of all even integers. Since A is unbounded, there is no limit to how large the positive integers can be. This means that B, which is a subset of A, is also unbounded.

4. Is it possible for B to be unbounded if A is bounded?

No, it is not possible for B to be unbounded if A is bounded. This is because B is a subset of A, so it cannot have a larger limit than A. If A is bounded, then B must also be bounded.

5. What implications does proving unboundedness of B if A is unbounded have?

If you are able to prove unboundedness of B if A is unbounded, then it means that B has no upper bound and can potentially grow infinitely. This can be useful in mathematical proofs and can have real-world applications in fields such as economics and physics.

Similar threads

I How do you define unboundedness in Euclidean space?
    • Calculus
Replies
4
Views
1K
I Unclear steps in a Zorich proof (Measurable sets and smooth mappings)
    • Calculus
Replies
2
Views
1K
A Lennard-Jones for bound/unbound atoms?
    • Atomic and Condensed Matter
Replies
1
Views
1K
I Can we physically realize unbound states?
    • Quantum Physics
Replies
16
Views
2K
B Proving half of the Heine-Borel theorem
    • Calculus
Replies
3
Views
1K
I Proving that an operator is unbounded
    • Linear and Abstract Algebra
Replies
25
Views
2K
I Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
    • Calculus
Replies
9
Views
846
I Discreteness of bound vs unbound states
    • Quantum Physics
Replies
11
Views
2K
MHB Heine-Borel Theorem in R^n .... Stromberg, Theorem 3.40 .... ....
    • Topology and Analysis
Replies
2
Views
1K
A Is a C^1 diffeomorphism with f of class C^k also a C^k diffeomorphism?
    • Calculus
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top