# Does the Supremum of the Set A Exist?

• silvermane
Your prof might not care about this (I mean asking you to prove that there is an upper bound). I am just telling you what is the right thing to do. Hope this helps.In summary, the supremum of the set A = {x:x in Q, x^3 < 2} is 2^(1/3). To prove that the supremum exists, it is necessary to show that the set is non-empty and bounded above. This can be done by demonstrating that 1 is an element of the set and that any number greater than 2 is not in the set. This proves that the set is bounded above and therefore, the supremum exists.
silvermane
Gold Member
Prove the supremum exists :)

## Homework Statement

Let A = {x:x in Q, x^3 < 2}.
Prove that sup A exists. Guess the value of sup A.

## The Attempt at a Solution

First we show that it is non-empty. We see that there is an element, 1 in the set, thus A is non-empty.
Now we show that A is bounded. We see that 2 is not in the set, thus there must be an upper bound on our set. Our set can be represented as (-oo, 2).
Since A is bounded above, then supA exists, and we are done.

(I just need help clarifying and making sure that I am following the correct logic here.)
=)

Also, would supA = 2?

I don't need an answer or anything, I just need someone to confirm that what I've shown here is a good proof.

Hey,

The suprema is not 2. :-)

It is the cubed root of 2.

You have to show that any number that is greater than ,lets says, 2 is not in the set.

Saying that 2 is not in the set is merely saying that two is not in the set it doesn't prove 2 is an upperbound. A little bit more is required to show the set is bounded above.

:-)

╔(σ_σ)╝ said:
Hey,

The suprema is not 2. :-)

It is the cubed root of 2.

You have to show that any number that is greater than ,lets says, 2 is not in the set.

Saying that 2 is not in the set is merely saying that two is not in the set it doesn't prove 2 is an upperbound. A little bit more is required to show the set is bounded above.

:-)

Awe... :/
That's how our professor did these problems in class. I'm so confused now >.<

Well the suprema you gave is incorrect. I don't think your prof did that in class .

Ex

S={0,1,2,5}
4 is not in the set and it is not an upper bound.

My point is that if you say something is an upperbound you have to show that it is greater than every value in the set.

In your example I can see it is obvious what is and what is not an upper bound but this is analysis; we have to show everything.

I guess your prof wanted you guys to fill in the details or he simply doesn't require very detailed solutions. So I guess you can get away with what you did .:-)

╔(σ_σ)╝ said:
Well the suprema you gave is incorrect. I don't think your prof did that in class .

Ex

S={0,1,2,5}
4 is not in the set and it is not an upper bound.

My point is that if you say something is an upperbound you have to show that it is greater than every value in the set.

In your example I can see it is obvious what is and what is not an upper bound but this is analysis; we have to show everything.

I guess your prof wanted you guys to fill in the details or he simply doesn't require very detailed solutions. So I guess you can get away with what you did .:-)

lol I should probably be more precise when I stated my work :(
We're using the continuum property and the fact that there are infinite elements in our set, and once we find one not in our set, then there's no other greater ones in the set.
In this example, he doesn't want a proof proof, I think he just wants us to say it's "obvious" with observation, and etc.

But say we were to prove it, for enlightenment, how would I go about it that way?

and yes, I believe I meant to say that 2^3 was the supremum, I was just being noodle-headed :3

silvermane said:
lol I should probably be more precise when I stated my work :(
We're using the continuum property and the fact that there are infinite elements in our set, and once we find one not in our set, then there's no other greater ones in the set.
In this example, he doesn't want a proof proof, I think he just wants us to say it's "obvious" with observation, and etc.

But say we were to prove it, for enlightenment, how would I go about it that way?

and yes, I believe I meant to say that 2^3 was the supremum, I was just being noodle-headed :3

You got the suprema wrong again .

It is 2^(1/3).

In the case you had to prove 2 is an upperbound you could start by using some axioms or simply state that for x<0, x^3 <0. So this automatically eliminates negative numbers as upper bounds.

Then you could show that x>2 , x^3 >8 which means it is not in the set.

Sorry, I am just been too picky. I guess we do not have to prove obvious things like these but I will be nice to at least state it without proof.

:-) The reason I am asking you to prove that there is an upper bound is that the question asked you to prove that the supremum exist. Which means you have to prove two things.
1) The set is non empty
2) It is bounded above.

If you were supposed to prove the suprema is x for example you can afford to wave your hand and say "clearly ... the set is bounded".
But when you are told to directly prove the suprema exist I think you need to prove those two thinks no matter how obvious it seems.

## 1. What does it mean for the supremum to exist?

The supremum (also known as the least upper bound) of a set is the smallest number that is greater than or equal to all the elements in the set. Therefore, proving that the supremum exists means showing that there is a specific number that satisfies this condition.

## 2. Why is proving the existence of the supremum important?

Proving that the supremum exists is important because it is a fundamental concept in real analysis and is used to define other important mathematical concepts, such as limits and continuity. It also allows us to make precise statements about the behavior of a set of numbers.

## 3. What is the difference between the supremum and the maximum of a set?

The supremum and the maximum of a set are both the largest number in the set, but they differ in how they are defined. The supremum is the smallest number that is greater than or equal to all the elements in the set, while the maximum is the largest number in the set. Not all sets have a maximum, but they all have a supremum.

## 4. How do you prove the existence of the supremum?

The most common way to prove the existence of the supremum is by using the completeness axiom. This axiom states that every nonempty set of real numbers that is bounded above has a supremum. Therefore, to prove the existence of the supremum, we need to show that the set is nonempty and bounded above.

## 5. Can the supremum of a set be a member of the set?

Yes, the supremum of a set can be a member of the set. For example, if we have the set {1, 2, 3}, the supremum is 3, which is also a member of the set. However, this is not always the case. For example, in the set (0,1), the supremum is 1, which is not a member of the set.

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