Proving SUP without a Formula: An Abstract Approach

[transgalactic]

i added the question in the link:

http://img388.imageshack.us/img388/2220/57946210jf5.gif

this is a total abstract thing

i don't know how to prove SUP whithout a formula to the series

??

 

[CompuChip]

What is the definition of supremum?

Also, you put a question mark with "least upper bound", do you know what it means?

 

[transgalactic]

i know the definition of supremum and least upper bound

i don't know what to do here because i don't have a formula for this series
??

 

[arildno]

i know the definition of supremum and least upper bound

i don't know what to do here because i don't have a formula for this series
??

It is NOT a question whether you know "the" definition of the supremum and least upper bound. I'm sure you do!

But:
After all, there are numerous definitions floating about, provably logically equivalent.

Thus, if we are to be able to help you, we need to know what particular definition YOU are to utilize in order to prove the statement; it is no good for us to prove it from a definition you are not entitled to use.

Therefore, please post the definitions you are allowed to use for this exercise.

 

[transgalactic]

my definitions of SUP and bound all came from this article:
https://www.math.purdue.edu/academic/files/courses/2007fall/MA301/MA301Ch6.pdf

 

[CompuChip]

They define sup as the least upper bound. But if I read your question correctly, they ask you to prove that "if S is the least upper bound of A and there is a sequence of elements in A converging to S, then S = sup(A)". So I must be missing something here, because that is trivial, isn't it?

 

[transgalactic]

sup is defined as a least upper bound in the question
if S exists in A then its defined as max(A)

 

[icantadd]

Half of the question seems unnecessary going by the definition of sup in the article. If S is the least upper bound for A, then S is synonymous with sup(A).

Definition 1. Let S be a set of real numbers. An upper bound for S is a number B such that x ≤ B for all x ∈ The supremum, if it exists, (“sup”, “LUB,” “least upper bound”) of S is the smallest upper bound for S.

Now if the question asks, assume there is a bounded set A, and you can find a sequence of numbers that converge to some value S, and S is in A and just an "upper bound", then it you can show it is the least upper bound.

 

[transgalactic]

sup=least upper bound

how to solve it?

 

[icantadd]

sup=least upper bound

how to solve it?

Done. Sup = least upper bound. Are you trying to prove a definition?

If M = sup S, the for every epsilon >0 there is an x in S, such that

M - epsilon < x <= M.

In other words, let x be as close to M as you want while still being in S, x will be less or equal to M.

To take this from another view, there are 12 axioms of a complete ordered field like R. One of them (#12 in my book) says that every non-empty set of real numbers S that is bounded from above has a least upper bound, called the sup S. I don't know what there is to solve/prove.

 

[CompuChip]

To be shown: If S is the least upper bound of A and there is a sequence of elements in A converging to S, then S = sup(A)

Proof: Let S be the least upper bound of A and assume there is a sequence of elements in A converging to S. Then in particular, S is the least upper bound of A. By definition 1, S is the supremum of A. QED.

 

[transgalactic]

i need to prove that S is the least upper bound
i don't know how to do that. i can say that
there is a cetain "e"
that if we subtract it from S we will get a member from A which is larger then S.

but is it a proof??

and what the -> thing means in the end ??

 

[CompuChip]

666 posts transgalactic... evil!

Anyway, I think the question is not entirely clear.
What do you want to prove and what are the assumptions?
Where did the link in your first post come from, did you scan it from a textbook or did you type it yourself? Why are some parts "pencil"-drawn? (Like $A \subseteq \mathbb{R}$, the question mark, ...)

Did you maybe mean: "Prove that S = sup(A) if S is an upper bound of A (not necessarily the least) and there is a sequence converging to S"?

 

[transgalactic]

i scan the questions from a textbook i am learning from

what is this arrow thing means?

 

[CompuChip]

It means that
$$\lim_{n \to \infty} a_n = S$$.