Proving Suprenum of A and B: Bob's Question

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To prove that sup(A ∪ B) = max(sup A, sup B), it is essential to establish that sup(A ∪ B) is an upper bound for both sets A and B. This involves verifying that sup A and sup B are indeed upper bounds for A ∪ B. The argument requires demonstrating that the maximum of these two suprema is the least upper bound for the union of the sets. Additionally, it is important to clarify that sup is not a set but an element of R, which affects how one approaches proving the properties of suprema. Understanding these definitions and relationships is crucial for successfully completing the proof.
Bob19
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Hello

I have two non-empty sets A and B which is bounded above by R.

Then I'm tasked with proving that

sup(A \cup B) = max(sup A, sup B)

which supposedly means that sup(A \cup B) is the largest of the two numbers sup A and sup B.

Can this then be written as sup(A) < sup(A \cup B) and sup(B) < sup(A \cup B) ?

Can this then be proven by showing that sup(A) < sup(A \cup B) is true?

Or am I totally on the wrong path here??

/Bob
 
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correct your tex and just verify the definitions of sup.
 
matt grime said:
correct your tex and just verify the definitions of sup.
My definition of Supremum is a follows:
every non-empty, bounded above subset of R has a smallest upper bound.

Then sup(A \cup B) has a larger smallest upper bound than sup(A) and sup(B) according to the definition of Supremum ?
Does this prove the given argument in my first post?
/Bob

p.s. If my idear is true, can this then be proven by taking a number z, which I then prove z \in sup(A \cup B) but z \notin sup(A) and z \notin sup(B) ?

/Bob
 
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why are you treating sup as a set (and taking elements in it?). Sup is not a set, it is an element of R.

Sup of a set is the least upper bound (when it exists)

obivously the least upper bound of AuB is the max of the least upper bounds, but you need to verify it, ie show it is an upper bound, and show it is the least upper bound. The first is easy, the second slightly harder.
 
matt grime said:
obivously the least upper bound of AuB is the max of the least upper bounds, but you need to verify it, ie show it is an upper bound, and show it is the least upper bound. The first is easy, the second slightly harder.


Okay those two aspects then prove the argument that

sup(AuB) = max(sup A, sup B) ?


/Bob
 
As I explained to someone else earlier tonight, I can easily see the answer becuase of experience, YOU need to demonstrate that you understand the answer by not having to let me fill in any blanks. If you don't see that an argument proves something then YOU need to do some work to rectify that, not me.
 

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