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Math Help - how to show xn>0 need not imply lin xn>a

  1. #1
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    how to show xn>0 need not imply lin xn>a

    Let (x_n) be a convergent sequence, and let a,b \in R.

    Show that for all n, x_n>a ned not imply limx_n>a.

    How do I show this when I have previously proven that If x_n>= a for all n, then lim x_n>=a. (this was another question that i had proven before hand) But how would I show it, when it contradicts the previously stated question.
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  2. #2
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    What about the sequence \displaystyle \left\{\frac{1}{n}\right\}. This is \displaystyle > -1 \forall n \in \mathbf{Z}^+ but \displaystyle \to 0 as \displaystyle n \to \infty
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  3. #3
    MHF Contributor chisigma's Avatar
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    The demonstration that \displaystyle \lim_{n \rightarrow \infty} x_{n} = a doesn't imply in any case that \forall n is  x_{n}>a or  x_{n}<a [the convergence may be 'oscillating'...] but that is \lim_{n \rightarrow \infty} |x_{n}-a|=0...

    Kind regards

    \chi \sigma
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