A problem on infimum and supremum. Prove.

Thread starter Kinetica
Start date Aug 3, 2011
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Supremum

In summary, infimum and supremum are important concepts in mathematics that help us define and understand the boundaries of a set. They allow us to prove the existence of certain elements in a set and make precise statements about the behavior of functions and sequences. To prove a problem involving infimum and supremum, one needs to use the definitions of these concepts and the properties of real numbers, and may also use other proof techniques such as mathematical induction. An example of such a problem is proving the infimum and supremum of a set. In real-world applications, understanding infimum and supremum can be useful in fields such as economics, physics, and engineering, as they can help optimize resources and predict the behavior of systems.

Aug 3, 2011

Kinetica

The problem is attached. So is the solution by contradiction.
I would love to hear your feedback if my sol. by contradiction is sufficient enough. Thanks a lot.

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Aug 4, 2011

lanedance

Homework Helper

so what are you trying to prove?

also i find you'll generally get a quicker response if you type or use tex, that way people can cut and paste and its easy to read...

1. What is the definition of infimum and supremum?

The infimum of a set is the greatest lower bound, meaning that it is the largest number that is still less than or equal to all the numbers in the set. The supremum of a set is the least upper bound, meaning that it is the smallest number that is still greater than or equal to all the numbers in the set.

2. Why is it important to understand infimum and supremum in mathematics?

Infimum and supremum are important concepts in mathematics because they help us define and understand the boundaries of a set. They also allow us to prove the existence of certain elements in a set and make precise statements about the behavior of functions and sequences.

3. How do you prove a problem involving infimum and supremum?

To prove a problem involving infimum and supremum, you need to use the definitions of these concepts and the properties of real numbers. You may also need to use mathematical induction or other proof techniques to establish the existence of certain elements in the set and the validity of the given statement.

4. Can you give an example of a problem on infimum and supremum?

One example of a problem on infimum and supremum is proving that the infimum of the set {(−1)n + 1/n | n ∈ N} is -2 and the supremum is 2. This can be proven by showing that -2 is the greatest lower bound and 2 is the least upper bound, and then using mathematical induction to prove that these bounds are indeed the infimum and supremum of the set.

5. How can understanding infimum and supremum help in real-world applications?

Infimum and supremum have many real-world applications, especially in economics, physics, and engineering. They can help us determine the minimum and maximum values of a function, optimize resources, and predict the behavior of systems. For example, in economics, infimum and supremum can be used to find the most profitable price for a product, while in physics, they can help determine the maximum and minimum possible values of physical quantities.