# Application of Supremum Property .... Garling, Theorem 3.1.1 ....

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In summary: A.This implies that m' is not an upper bound for A, and there must be an element in A that is larger than m'. Let's call this element m''. By the same reasoning as above, we can show that m'' is not an element of A, and there must be an element in A that is larger than m''. Continuing this process, we can keep finding elements in A that are larger than the previous one, which means that A is infinite.Therefore, we have shown that A cannot be infinite, and it must be finite. This logic is what is used to prove the finiteness of A using (ii), and this completes the proof of The
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I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ...

I am focused on Chapter 3: Convergent Sequences

I need some help to fully understand the proof of Theorem 3.1.1 ...Garling's statement and proof of Theorem 3.1.1 (together with some interesting remarks) reads as follows:
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View attachment 9027
In the above text by Garling, in the proof of statement (iii), we read the following:

" ... ... Let $$\displaystyle A = \{ k \in \mathbb{Z}^{ + } \ : \ k \leq nx \}$$. A is non-empty ( $$\displaystyle 0 \in A$$ ) and finite, by (ii) ... ... " My question is as follows:

Although it is plausible (maybe even "obvious" ... ) that A is finite ... how do we logically and rigorously prove this using (ii} ... that is, logically and rigorously, how does (ii) imply that A is finite ...?

Help will be appreciated ... ...

Peter

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Hello Peter,

Thank you for reaching out for help with understanding the proof of Theorem 3.1.1 in Garling's book. I can see why you are having trouble understanding the implication of (ii) in proving that A is finite. Let me try to explain it in a more detailed and step-by-step manner.

First, let's recall what statement (ii) in the theorem states:

(ii) If A is a non-empty subset of \mathbb{Z}^{ + } such that A is bounded above, then A has a maximum element.

Now, let's look at the set A defined in the proof of statement (iii):

A = \{ k \in \mathbb{Z}^{ + } \ : \ k \leq nx \}

From this definition, we can see that A is a subset of \mathbb{Z}^{ + }, the set of positive integers. Also, A is bounded above by nx, since all elements in A are less than or equal to nx. Therefore, according to (ii), A has a maximum element, say m. This means that m is the largest element in A, and any other element in A is less than or equal to m.

Now, let's consider the set B = \{ k \in \mathbb{Z}^{ + } \ : \ k \leq (n+1)x \}. This set is also bounded above by (n+1)x, and according to (ii), it has a maximum element, say m'. Similar to m, m' is the largest element in B, and any other element in B is less than or equal to m'.

But notice that A is a subset of B, because any element in A is also in B (since A is defined as the set of all positive integers less than or equal to nx, and B is defined as the set of all positive integers less than or equal to (n+1)x). Therefore, m' is also an upper bound for A, and since m is the maximum element of A, we must have m' \geq m.

However, this means that m' is also an element of A, since m' is the largest element in B, and any other element in B is less than or equal to m'. But this contradicts the fact that m is the maximum element of A. Therefore, our assumption that m' is an element of A

## 1. What is the Supremum Property and how is it applied in mathematics?

The Supremum Property, also known as the Least Upper Bound Property, is a fundamental concept in mathematics that states that every non-empty set of real numbers that is bounded above has a least upper bound. This means that for any set of real numbers, there is a largest number that is still smaller than any other number in the set. This property is used in various mathematical proofs and theorems, including Theorem 3.1.1 in Garling's book.

## 2. Can you explain Theorem 3.1.1 in Garling's book and its significance?

Theorem 3.1.1 in Garling's book states that if a set of real numbers has a supremum, then it is also the limit of a sequence of numbers in that set. This theorem is significant because it allows us to prove the existence of limits for certain functions and sequences, which is crucial in many areas of mathematics, including analysis and calculus.

## 3. How is the Supremum Property used in real-world applications?

The Supremum Property is used in various real-world applications, including economics, finance, and optimization problems. For example, in economics, it is used to determine the maximum price that a consumer is willing to pay for a certain good or service. In finance, it is used to determine the maximum profit or minimum loss for a particular investment. In optimization problems, it is used to find the best solution among a set of options.

## 4. Are there any limitations or exceptions to the Supremum Property?

While the Supremum Property holds true for most sets of real numbers, there are some exceptions and limitations. For example, it does not hold for sets of complex numbers or for infinite sets. Additionally, there are certain sets of real numbers that do not have a supremum, such as the set of all rational numbers.

## 5. How can I apply the Supremum Property in my own mathematical proofs and problems?

The Supremum Property is a useful tool in many mathematical proofs and problems, particularly in the areas of analysis and calculus. To apply it, you must first identify if the set of numbers in question is bounded above. If it is, then you can use the Supremum Property to prove the existence of a limit or to find the maximum or minimum value in the set. It is important to carefully consider the properties and limitations of the Supremum Property when using it in your own work.

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