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Dedekind Cut and Archemedean Q

  1. Mar 9, 2014 #1
    Let α be a Dedekind Cut. w a positive rational.How to prove that there exists a integer n such that nw is a member of α and (n+1)w is not a member of α, using Archemedian propoerty of Q.

    Suppose p is a member of α. we can find n such that nw < p < (n+1)w. So nw is
    a member of α. Further I am not able to proceed.

    Please help me.
     
    Last edited: Mar 9, 2014
  2. jcsd
  3. Mar 9, 2014 #2
    OK. Looks like I got the answer. It was a trivial case. If there is no such n then nw is a member of alpha for every n by induction. Since aplpha is not equal to Q and Q is archemedian it is not possible.If q is not a member
    of alpha then q > any member of alpha and nw > q for some n.And nw cannot be a member of alpha.

    Sporry for asking such a simple question.
     
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