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

AutGuy98

New member
Sep 11, 2019
20
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

Well-known member
MHB Math Scholar
Oct 18, 2017
247
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 $\sup A = \sup B = 1$.