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Math Help - Archimedean property

  1. #1
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    Archimedean property

    So there is this statement, claiming that N (Natural numbers) is not bounded above in R, which is really easy to prove, however after proving that they drew a conclusion as I saw in my lecture notes,

    Conclusion; For every epsilon (I will write epsilon as e because I don't know how else to write it) greater than zero there exist n in N with 0<1/n<e

    How did he drew this conclusion from the fact that N is not bounded above in R? From what I understand what this says , for every epsilon in R (that is greater than 0) you can find a natural number n that is bigger than him. Is this logic correct?
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  2. #2
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    Re: Archimedean property

    Quote Originally Posted by davidciprut View Post
    Conclusion; For every epsilon (I will write epsilon as e because I don't know how else to write it) greater than zero there exist n in N with 0<1/n<e How did he drew this conclusion from the fact that N is not bounded above in R?
    Using LaTeX you can enter [TEX]\forall\varepsilon>0 [/TEX] gives \forall\varepsilon>0

    If \mathbb{N} is not bounded above then \frac{1}{\varepsilon} is not an upper bound.
    So \exists n\in\mathbb{N} such that n>\frac{1}{\varepsilon} . Can you see how it works?
    Last edited by Plato; January 31st 2014 at 12:25 PM.
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