Thread: real analysis supremum question

1. real analysis supremum question

I have another question. Would someone mind showing me how to do this? Let S be a bounded nonempty subset of the real numbers. Let a and b be fixed real numbers. Define the set T as {as+b: s in S}. Find formulae for supT and infT in terms of supS and supT and prove your formulae.
Thank you for any help and thanks for help from before.

2. Originally Posted by BrainMan
Find formulae for supT and infT in terms of supS and supT and prove your formulae.
please check to make sure you have the correct things where you see red.

3. Use the following facts for non-empty sets.

$\sup T = \sup S + a$ where $T = \{ x + a| x\in S \}$.

$\sup T = b\sup S$ where $T = \{ bx| x\in S \} , \ b>0$.

$\sup T = b\inf S$ where $T = \{bx| x\in S \} , \ b<0$.

4. Yeah, so it should read, "Find formulae for supT and infT in terms of supS and infS and prove your formulae."
Sorry about that.
PerfectHacker: What are you saying? Could you clarify?

5. Originally Posted by BrainMan
Yeah, so it should read, "Find formulae for supT and infT in terms of supS and infS and prove your formulae."
Sorry about that.
PerfectHacker: What are you saying? Could you clarify?
the elements of the set T are given by as + b for $s \in S$. when are the elements of T the smallest? when are they the largest?

6. How many different combinations are there? I'm thinking I only need to find one formula for each of the supT and infT. Yet, it appears the formulae are different depending on whether a is positive or negative, etc. I'll probably be able to prove whatever the answer is, but can you just lead me in the right direction to actually get the formula?
It seems for a>0, supT= asupS+b and infT= ainfS+b. For a<0, supT= ainfS+b and infT=asupS+b.

7. And then this is affected if the smallest element in S is negative. I don't know. I'm just confused.

8. So I did some stuff out and I think this is right. Can anyone confirm?
If a is positive or zero, then supT=asupS+b and infT=ainfS+b.
If a is negative, then supT=ainfS+b and infT=asupS+b.

9. Originally Posted by BrainMan
So I did some stuff out and I think this is right. Can anyone confirm?
If a is positive or zero, then supT=asupS+b and infT=ainfS+b.
If a is negative, then supT=ainfS+b and infT=asupS+b.
looks ok

10. In an analysis type course, do you think you need to use epsilon notation to prove these?
Is it ok just to say that supS is a supremum and multiplying it by a constant a and adding a constant b (the definition of the set T) will make this the supremum also?

11. Originally Posted by BrainMan
So I did some stuff out and I think this is right. Can anyone confirm?
If a is positive or zero, then supT=asupS+b and infT=ainfS+b.
If $a=0$ then you need to be careful. For the set $S$ can be unbounded.
It is correct to write,
$\sup T = b$ in that case.

12. Originally Posted by ThePerfectHacker
If $a=0$ then you need to be careful. For the set $S$ can be unbounded.
It is correct to write,
$\sup T = b$ in that case.
we were told S was bounded, so it's ok

13. Can someone help me prove this? I'm kind of struggling.

14. Originally Posted by BrainMan
Can someone help me prove this? I'm kind of struggling.
Prove what?

15. Prove this: If a is positive or zero, then supT=asupS+b and infT=ainfS+b.
If a is negative, then supT=ainfS+b and infT=asupS+b.

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