# Proving the Uniqueness of inf(A) in a Subset of ℝ | Epsilon Proof Technique

• jaqueh
In summary, when attempting to prove that the infimum of a subset A of the real numbers is unique, one can start by assuming that there are two infimums x and y for A and then showing that x and y must be equal by using the definition of infimum as a greatest lower bound. Alternatively, one could use an epsilon proof to show that |x-y| is less than a certain value, proving that x and y must be equal. The specific method used will depend on the definition of infimum being used.
jaqueh

## Homework Statement

Let A be a subset of ℝ, now prove if inf(A) exists, then inf(A) is unique

## The Attempt at a Solution

I am using an epsilon proof but i don't think i am going about it the right way, can someone nudge me in the right direction

jaqueh said:

## Homework Statement

Let A be a subset of ℝ, now prove if inf(A) exists, then inf(A) is unique

## The Attempt at a Solution

I am using an epsilon proof but i don't think i am going about it the right way, can someone nudge me in the right direction

How you do it depends very much on what *definition* you are using for "inf". Different (but equivalent) definitions would need different types of proofs.

RGV

Usually, for something like this, it is best to start of saying "Let x and y be inf(S)" and then show that x and y are, in fact, the same. Now, showing that |x - y| < epsilon might be a good strategy, but there are probably more efficient methods. But, as RGV said, it depends on how inf is defined for you. If you have defined it as a greatest lower bound, the method I mentioned works very well.

yes i think I am going to revise it as x and y are infimums for A then x<y and y<x so x=y

jaqueh said:
yes i think I am going to revise it as x and y are infimums for A then x<y and y<x so x=y

That's a contradiction the way you wrote it. Did you mean $x \leq y, \ \text{and} \ y \leq x,\ \text{so} \ x=y$?

yes that is what i meant and i did write it that way x≥y y≥x

## 1. What is the "inf(A)" in the context of proving uniqueness in a subset of real numbers using the epsilon proof technique?

The "inf(A)" refers to the infimum, or greatest lower bound, of a set of real numbers A. In other words, it is the smallest number that is greater than or equal to every element in A.

## 2. How is the uniqueness of inf(A) proven using the epsilon proof technique?

The epsilon proof technique involves showing that for any given positive number epsilon, there is only one possible value for inf(A). This is done by assuming that there are two different values for inf(A) and then using the properties of the infimum to show that this leads to a contradiction.

## 3. Why is it important to prove the uniqueness of inf(A) in a subset of real numbers?

Proving the uniqueness of inf(A) is important because it allows us to make precise statements about the behavior of functions and sets of real numbers. It also helps us to establish important properties of real numbers, such as the completeness property.

## 4. What are some common difficulties when using the epsilon proof technique to prove the uniqueness of inf(A)?

One common difficulty is finding the right value for epsilon and showing that it leads to a contradiction. Another difficulty is understanding the properties of the infimum and how they can be used to prove uniqueness.

## 5. Can the epsilon proof technique be used to prove uniqueness of other mathematical concepts besides inf(A)?

Yes, the epsilon proof technique can be used to prove uniqueness in a variety of mathematical contexts, such as the supremum of a set or the limit of a sequence. It is a powerful tool in mathematical analysis and can be applied to many different situations.

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