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Completeness axiom/theorem and supremum

  1. Dec 11, 2005 #1
    "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.
     
  2. jcsd
  3. Dec 11, 2005 #2

    HallsofIvy

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    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?
     
  4. Dec 11, 2005 #3

    matt grime

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    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.
     
  5. Dec 11, 2005 #4
    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=(????).
     
  6. Dec 11, 2005 #5

    matt grime

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    we may suppose e<z and e<y, can you see how that might help?
     
  7. Dec 11, 2005 #6
    e>0. z and y could be negative.
     
  8. Dec 12, 2005 #7

    matt grime

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    How can the supremum of a set of positive numbers be negative?
     
  9. Dec 12, 2005 #8
    A and B aren't sets of strictly positive numbers.
     
  10. Dec 12, 2005 #9

    matt grime

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    Then the proposition is trivially false.
     
  11. Dec 12, 2005 #10
    You're right. This is odd considering it's an archived analysis final exam.
     
  12. Dec 12, 2005 #11

    matt grime

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    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.
     
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