Does this count as a proof for the infimum?

In summary, the infimum of the set is proven to be by showing that it is the smallest lower bound for the set. This is done by defining the concept of infimum on a partially ordered set and using an proof to show that is indeed the infimum of the set.
  • #1
NoName3
25
0
I want to know whether the following the counts as a proof that infimum of the set is .

Let , where is some ordered field. Then is such that for any we have and for any number such that we have . I'm not sure as to whether what's below misses the second part of the definition. So, is it enough?

Let , then and .

Let then Thus Hence is strictly decreasing from to . Similarly, let then Hence is strictly decreasing from to . Hence we have which implies that . Thus Is this correct, do I've to use proof?
 
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  • #2
NoName said:
Let , where is some ordered field.
The concept of infimum is defined on any (partially) ordered set.

NoName said:
Then is such that for any we have and for any number such that we have .
This is not stated very well. The scope of the universally quantified is the clause "we have ". After that, the scope closes and is no longer defined. Therefore, the second occurrence of , namely, in the phrase "such that ", requires its own universal quantifier. Note also that the scope of this second quantifier does not last until the end of the sentence, but only includes "". Formally,

is incorrect because the occurrence of in is undefined. Also,

is incorrect because it has a different meaning. The correct statement is

In words, it says, " is such that for any we have and for any number such that for all we have ".

The rest of the proof also has some flaws, so I'll write my variant. We have to prove and . For the first claim,

For the second one, suppose that for all , but . Let and consider an odd . Then

Therefore,

We found an such that , which contradicts the assumption that for all .
 
  • #3
Thank you, very helpful!
 

FAQ: Does this count as a proof for the infimum?

What is the definition of infimum?

The infimum of a set is the greatest lower bound, or the largest number that is less than or equal to all numbers in the set.

How is the infimum related to proofs?

The infimum is often used in mathematical proofs to show that a certain number is the smallest possible value in a given set or function.

Can any number be considered an infimum?

No, not all numbers can be an infimum. For a number to be an infimum, it must be less than or equal to all numbers in the set and there cannot be a smaller number that also meets this criteria.

What is the difference between infimum and minimum?

The infimum is the greatest lower bound of a set, while the minimum is the smallest value in the set. The minimum is always a member of the set, while the infimum may or may not be.

How do you prove that a number is the infimum?

To prove that a number is the infimum, you must show that it is less than or equal to all numbers in the set and that there is no smaller number that also meets this criteria. This can be done through various mathematical techniques such as contradiction, induction, or direct proof.

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