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Math Help - i have proved this, does it imply boundedness?

  1. #1
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    i have proved this, does it imply boundedness?

    f : X --> R

    I proved: \exists M\geq 0, \forall \epsilon>0, \forall x\in X, M-\epsilon < |f(x)|< M+\epsilon
    Last edited by szpengchao; October 19th 2008 at 07:05 AM.
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  2. #2
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    Quote Originally Posted by szpengchao View Post
    f : X --> R

    I proved: \exists M\geq 0, \forall \epsilon>0, \forall x\in X, M-\epsilon < |f(x)|< M+\epsilon
    I think you actually probably proved M - \epsilon < f(x) < M + \epsilon for x\in X.

    Yes, this proves that f is bounded on X.
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  3. #3
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    Quote Originally Posted by szpengchao View Post
    f : X --> R

    I proved: \exists M\geq 0, \forall \epsilon>0, \forall x\in X, M-\epsilon < |f(x)|< M+\epsilon
    If you can show that \exists M\geq 0, \forall \epsilon>0, \forall x\in X,  |f(x)|< M+\epsilon
    then it follows that \forall x\in X, |f(x)|\le M or f is bounded on X.
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