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Math Help - Countability problem

  1. #1
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    Countability problem

    I've tried for a long time, still can't figure it out. Please help me, thanks.

    Let T be a nonempty subset of (−1, 0)union(0, 1).
    If every finite subset {x1, x2, . . . ,xn} of T (with no two of x1, x2, . . . ,xn equal) has the property that x1^2+x2^2+ +xn^2 < 1, then prove that T is a countable set.
    (Hint: For every positive integer k, is the set T_k = {x : x belongs to T and |x| belongs to [1/(k+1),1/k)} a finite set? What is the union of [1/(k+1), 1/k) for k = 1, 2, 3, . . ..)
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  2. #2
    Super Member girdav's Avatar
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    Re: Countability problem

    I guess you tried to use the hint: where are you stuck?
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  3. #3
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    Re: Countability problem

    In fact, I'm ok with the hint part. But I can't figure out how to prove T_k as a finite set.
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  4. #4
    Super Member girdav's Avatar
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    Re: Countability problem

    If x\in T, then |x|\in (0,1) so |x|\in ((k+1)^{-1},k^{-1}) for some k. This gives that x\in T_k. Conversely, if x is in a T_k, it's in T.
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  5. #5
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    Re: Countability problem

    First, really thanks for your kindness help.
    What I need is the part that utilize #x1^2+x2^2+ +xn^2 < 1# to prove T_k is finite set, I just can't link them up.
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  6. #6
    Super Member girdav's Avatar
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    Re: Countability problem

    What if there were infinitely many elements in T_k (for example, more than k+2)?
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  7. #7
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    Re: Countability problem

    Since |x| greater than or equal to 1/(k+1), there is at most (k+1)^2 elements, otherwise it'll contradict the inequality.
    Is it correct?
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  8. #8
    Super Member girdav's Avatar
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    Re: Countability problem

    Yes.
    Thanks from wesleybrown
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  9. #9
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    Re: Countability problem

    Haha, solved. That's great. Nice to meet you, thank you very much!
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