Completeness axiom/theorem and supremum

[Treadstone 71]

"Let A and B be bounded nontempty sets of real numbers. Let C={ab:a in A, b in B}. Prove that sup(C)=sup(A)sup(B)."

Here's what I've done so far:

By the completeness axiom/theorem A and B have suprema. Let sup(A)=z and sup(B)=y. For all e>0, there exists a in A and b in B such that z-e<a and y-e<b. Multiplying the two inequalities and we have

zy-ze-ye+e^2 < ab

I'm stuck here.

 

[HallsofIvy]

"Let A and B be bounded nontempty sets of real numbers. Let C={ab:a in A, b in B}. Prove that sup(C)=sup(A)sup(B)."
Here's what I've done so far:
By the completeness axiom/theorem A and B have suprema. Let sup(A)=z and sup(B)=y. For all e>0, there exists a in A and b in B such that z-e<a and y-e<b. Multiplying the two inequalities and we have
zy-ze-ye+e^2 < ab
I'm stuck here.

It would be a good idea to show first that zy is an upper bound on C!
You are using the fact that since z is the least upper bound on A there must be an a between z-e and z and since y is the least upper bound on B there must be a b between z-e and z. What does that tell you about there being an ab between zy- e and z?

 

[matt grime]

step 1 show it's an upper bound,

step 2 show it is a least upper bound which can be messy.

it's easier to use the definition of sup as the maximum of the accumulation points.

You need to show that give d>0 you can find e greater than zero such that 0<ey+ez-e^2<d, so do that.

 

[Treadstone 71]

I don't know, but I want to arrive at the conclusion that zy-x<ab for all x>0. If I could prove that zy-(?)<ab where (?) is positive, then I'm done, since I can let x=(?).

 

[matt grime]

we may suppose e<z and e<y, can you see how that might help?

 

[Treadstone 71]

e>0. z and y could be negative.

 

[matt grime]

How can the supremum of a set of positive numbers be negative?

 

[Treadstone 71]

A and B aren't sets of strictly positive numbers.

 

[matt grime]

Then the proposition is trivially false.

 

[Treadstone 71]

You're right. This is odd considering it's an archived analysis final exam.

 

[matt grime]

The odd thing is that at one point i even checked back to make sure that these were sets of positive numbers just so i didn't make a mistake and i am convinced that i remember reading that you wrote they were positive real numbers. Odd.