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Math Beauties need HELP, part II

  1. Oct 6, 2007 #1
    1. The problem statement, all variables and given/known data

    Let S be the nonempty set of real numbers bounded above. Prove that S^3 = {x^3 : x [tex]\in[/tex] S} is bounded above and sup S^3 = (sup S)^3

    2. Relevant equations

    given S^3 = { y [tex]\in[/tex] R : [tex]\exists[/tex] x, x [tex]\in[/tex] S and y = x^3}

    and

    for all [tex]\epsilon[/tex] > 0, there is y [tex]\in[/tex] S^3 such that [tex]\alpha[/tex]^3 < y [tex]\leq[/tex] [tex]\alpha[/tex]^3.

    3. The attempt at a solution

    This is what we attempted, but were told we are wrong:

    Let S= { s1, s2, s3,....} s.t. s1 > s2 > s3 > ...
    Then s1 [tex]\geq[/tex] sn, for all sn [tex]\in[/tex] S.
    This implies S is bounded above by S1 and so supS = s1

    Now:
    (supS)^3 = (s1)^3

    if s1 is negative, then (s1)^3 = (-s1)(-s1)(-s1) = -s1^3
    if s1 is positive, then (s1)^3 = (s1)(s1)(s1) = s1^3
    which implies that (supS)^3 = s1^3

    For S^3 = {s1^3, s2^3, s3^3,...} and s1^3 > s2^3 > s3^3 >...
    Then s1^3 [tex]\geq[/tex] sn^3, for all sn^3 [tex]\in[/tex] S^3.
    This implies S^3 is bounded above by S1^3 and so supS^3 = s1^3

    therefore, supS^3 = (supS)^3 = s1^3
     
  2. jcsd
  3. Oct 6, 2007 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Well, that, to start with, makes no sense. I assume you mean a nonempty set of real numbers bounded above

    Are you assuming the set is countable? That's the only way you could write them like this. And sets of real numbers, in general, are not countable.[/quote]
    Then s1 [tex]\geq[/tex] sn, for all sn [tex]\in[/tex] S.
    This implies S is bounded above by S1 and so supS = s1[/quote]
    But you are also assuming that s1 is IN S and you were not told that sup(S) was in S.
    "sup(S)" is the least upper bound of S. Assuming x is in S3, then x= s3 for some s in S and so [itex]s\le sup(S)[/itex]. Can you then prove that [itex]x= s^3\le (sup(S))^3[/itex] (so that (sup(S))3 is an upper bound on S3)? Can you now prove that (sup(S))3 is the LEAST upper bound?

     
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