# Proof of the least upper bound

• furi0n
In summary, the question is asking to prove that if a set S of real numbers has a maximum value x0, then x0 must also be the least upper bound (sup) of the set. This can be proven by assuming that there exists another upper bound a that is less than x0, which leads to a contradiction. This question serves to test the understanding of the definition of least upper bound.
furi0n

## Homework Statement

LEt S is supset of real numbers and suppose that there is X0 is member of S such that x0>=x for all x which is member of S(i.e. x0 is the maximum of S). show that x0=supS

## The Attempt at a Solution

Not: this seems too easy question but i can't understand how ı can prove it please help me it's my important homework.. :(

Well x_0 is the maximum of x. Suppose a was an upper bound of S but a was not equal to x_0.

The a>x_0 right ?

Thus is there any upper bound smaller than x_0 ?

So we define x_0 as a value such that x_0>=x for all x in S. Now, suppose x_0 was not the sup of S (i.e. x_0 is not equal to supS). Well then this must imply the existence of a value b in S such that x_0 < b <= supS. Notice the contradiction? how did we define x_0 again?

Yes, it is an easy question! Since $x_0\ge x$ for all x in the set, it is an upper bound on the set. Is it possible for any other upper bound to be less than $x_0$? No, because $x_0$ is in the set and saying $M< x_0$ would contradict the definition of "upper bound".

thank you everybody, why i ask for this question is to learn how to prove this because we know already everything there is not anything which we can prove but today İ understood Teacher asked this Question in order to understand whether we learn to definition of least upper bound :D thank you again.

## 1. What is "Proof of the least upper bound"?

"Proof of the least upper bound" is a mathematical concept that is used to prove the existence of a certain number in a set that is greater than or equal to all other numbers in the set. It is also known as the "least upper bound property" or the "completeness property."

## 2. Why is "Proof of the least upper bound" important?

"Proof of the least upper bound" is important because it provides a way to guarantee the existence of a certain number in a set, even when that number is not explicitly stated. This is particularly useful in calculus and analysis, where it allows for the use of the supremum and infimum to solve problems.

## 3. How is "Proof of the least upper bound" used in real life?

"Proof of the least upper bound" is used in various fields, such as economics, engineering, and physics, to find the best possible solution to a problem. For example, in economics, it can be used to determine the maximum price a consumer is willing to pay, while in engineering, it can be used to determine the maximum weight a bridge can support.

## 4. What is the difference between "Proof of the least upper bound" and "proof by contradiction"?

The main difference between "Proof of the least upper bound" and "proof by contradiction" is the approach used to prove a statement. "Proof of the least upper bound" uses the least upper bound property to show the existence of a certain number, while "proof by contradiction" assumes the opposite of the statement and shows that it leads to a contradiction, thus proving the statement to be true.

## 5. Are there any limitations to "Proof of the least upper bound"?

Yes, there are limitations to "Proof of the least upper bound." This concept only applies to sets that have an upper bound and cannot be used for sets that do not have a highest element. Additionally, it does not guarantee the uniqueness of the least upper bound, meaning there may be more than one number that satisfies the property.

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