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Math Help - Real Analysis: delta epsilon

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    Real Analysis: delta epsilon

    Ok, to be honest, I have no idea where to start for this problem, and I',m not sure to to annotate some of it:

    Let f: D-->R and let c be an accumulation point of D. Assume lim f(x) = L, L>0
    x->c

    Prove using the epsilon- delta argument there exists a deleted neighborhood N*(c,epsilon) so that f(x) >0 for all xEN*(x, epsilon) (intersection) D.


    That's all that was given, I have a final exam on Friday, any help would be greatly appreciated.
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    Quote Originally Posted by Vitava61 View Post
    Ok, to be honest, I have no idea where to start for this problem, and I',m not sure to to annotate some of it:

    Let f: D-->R and let c be an accumulation point of D. Assume lim f(x) = L, L>0
    x->c

    Prove using the epsilon- delta argument there exists a deleted neighborhood N*(c,epsilon) so that f(x) >0 for all xEN*(x, epsilon) (intersection) D.


    That's all that was given, I have a final exam on Friday, any help would be greatly appreciated.

    Since the limit exists at c, we can find a delta for every epsilon

    \exists \delta > 0, \ni \mbox{ } |x-c|< \delta, |f(x)-L|< \epsilon

    Let \epsilon=L so we can find a \delta
    such that when
    |x-c|<\delta |f(x)-L|<\epsilon=L

    Now working on the 2nd we have

    -L < f(x)-L< L \iff 0< f(x) < 2L

    By the above when x \in (c-\delta,c+\delta)

    f(x) > 0

    QED
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