How Do You Prove That sup(S ∪ T) Equals max{supS, supT}?

[???TRU???]

Please help with analysis proof!

hey all,

i was wondering if anyone could offer some advice on how to complete or even begin this proof. here it is:

prove that sup(S union T)= max{supS,supT}. Do not assume S is a subset of T.


AHHHHHH!

thanks.

 

[matt grime]

step by step: what is the definition of sup{SuT}?

 

[mathwonk]

this is trivial. you have not even tried it have you?

i.e. intuitively, the biggest number in the union of two sets is either the biggest number in one or the other.

this problem is the same plus the clumsiness of sups.

 

[???TRU???]

this is trivial. you have not even tried it have you?

i.e. intuitively, the biggest number in the union of two sets is either the biggest number in one or the other.

this problem is the same plus the clumsiness of sups.

no, no I've tried. I understand the answer, the problem for me is showing it. Mainly that the max of the set exists in the union.

i guess another problem is i can't get around SupT doesn't have to exist in Sup SUT, but it could be the max of the set.

Oh well.

p.s. what's with the rage? I am not claiming to be an expert.


thanks to you two for replying.

 

[mathwonk]

It'ds not rage its frustration. sorry. start as suggested with the definition of sup. L is the sup of a set S if no element of the set S is larger than L, but elements of S do get as close to L as desired. Hence if L is the larger of the sups of S and T then L is at least as large as all elements of both S and T, and if L is the sup of S say, then surely there are elements of S hence also of SuT which get as close to L as you want. hence L is alkso the sup of SuT.

Where did you get stuck here? Thjs follows immediately from the definitions of the words.


now you try the other direction.