Proof of the Equality of Supremums (Or Something Like That Anyway :) )

[AutGuy98]

Hey guys,

I have an Intermediate Analysis problem that needs assistance. I've really been having a hard time with it. This is what the question says:

"Can it happen that A⊂B (A is a subset of B) and A≠B (A does not equal B), yet sup A=sup B (the supremum of A equals the supremum of B)? If so, give an example. If not, prove why not."

Honestly, I'm not even sure where to begin with proving this, so any help would be greatly appreciated on my behalf. Thank you in advance to anyone that replies.

 

[castor28]

Hey guys,

I have an Intermediate Analysis problem that needs assistance. I've really been having a hard time with it. This is what the question says:

"Can it happen that A⊂B (A is a subset of B) and A≠B (A does not equal B), yet sup A=sup B (the supremum of A equals the supremum of B)? If so, give an example. If not, prove why not."

Honestly, I'm not even sure where to begin with proving this, so any help would be greatly appreciated on my behalf. Thank you in advance to anyone that replies.

Hi AutGuy98,

Here is a simple example: take $A=\{1\}$ and $B=\{0,1\}$. You have $\sup A = \sup B = 1$.

 

[soloenergy]

Hi there,

I can definitely understand your struggle with this problem. It can be tricky to wrap your head around at first, but I'll do my best to explain it to you.

To answer the question, yes, it is possible for A to be a subset of B and for A to not equal B, yet have the supremum of A equal the supremum of B. Here's an example:

Let A = {1, 2, 3} and B = {1, 2, 3, 4}

In this case, A is definitely a subset of B because all the elements in A are also in B. However, A does not equal B because B has an additional element, 4, that is not in A. But if we look at the supremum of both A and B, we can see that they are both equal to 3, since that is the highest value in both sets.

Now, to prove why this is the case, we need to understand what the supremum of a set is. The supremum, or least upper bound, of a set is the smallest number that is greater than or equal to all the numbers in the set. In our example, 3 is the smallest number that is greater than or equal to all the numbers in both A and B.

So, even though A and B are not equal sets, they can still have the same supremum because the supremum is not determined by the exact elements in the set, but rather by the upper bound of those elements.

I hope this helps you understand the concept better. Let me know if you have any other questions or need further clarification. Good luck with your problem!