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Math Help - Bolzano’s theorem and the Archimedian property of R.

  1. #1
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    Bolzano’s theorem and the Archimedian property of R.

    Use Bolzano’s theorem and the supremum to prove the Archimedian property of R.
    I know bolzano's theorem shows that supA exists and the Arc.property of R shows that n>x.. but how do I use those two methods to prove one another?
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  2. #2
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    Could you describe the Bolzano’s theorem and the Archimedian property more precisely? This is not Bolzano–Weierstrass theorem, is it? Or is it the extreme value theorem?
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  3. #3
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    A⊂R is bounded, A≠0, and A⊂R is bounded above, SupA exists.
    This is proven by bisection.
    Thats all I have on his theorem
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  4. #4
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    The archimedian property is: For any x, there exists a natural number n with n>x.

    You can prove that R has the archimedian property this way:
    Let x>0 and
    A={n : n is a natural number and n<=x}
    Then A is bounded above by x and 0\in A,
    so sup A=s exists.
    Since s is the least upper bound of A, s-1 is not an upper bound of A anymore.
    So there exists n \in A with s-1<n
    =>
    s<n+1
    Since s is an upper bound of A
    => n+1 \notin A
    =>
    x<n+1

    i.e. m:=n+1 is a natural number with m>x.
    qed
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