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Help Me Understand the Archimedean Property

  1. Sep 13, 2011 #1
    This isn't really hw. I need someone to explain a certain line in a proof:

    " b2 [itex]\leq[/itex] [itex]\frac{1}{n}[/itex] for all n in the natural numbers. This implies that b2 [itex]\leq[/itex] 0 (a consequence of the Archimedean property). "

    I don't see how the Archimedean is applied in this context. This is my understanding of the property: For real numbers x,y, x>0, there exists a natural number n such that nx > y.

    I don't see how this proves b2 [itex]\leq[/itex] [itex]\frac{1}{n}[/itex] [itex]\Rightarrow[/itex] b2 [itex]\leq[/itex] 0.

    Thanks
     
  2. jcsd
  3. Sep 13, 2011 #2

    Dick

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    Apply the Archimedean property with b^2=x and y=1. You know b^2>=0. If b is not zero then b^2>0. If nb^2>1 then b^2>1/n. It's a proof by contradiction.
     
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