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Math Help - show there is exactly one n such that n<x<n+1

  1. #1
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    show there is exactly one n such that n<x<n+1

    If x \notin Z (the integers) show that there is exactly one n \inZ such that n<x<n+1.
    Would I show this by contadiction stating x \inZ?
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  2. #2
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    We can suppose that x>0 first.
    We know that \left( {\exists n \in \mathbb{Z}} \right)\left[ {x < n} \right]. By well ordering let K be the least to have that property.
    What can be said about K-1?

    If x<0 then do it for -x.
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  3. #3
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    Quote Originally Posted by Plato View Post
    We can suppose that x>0 first.
    We know that \left( {\exists n \in \mathbb{Z}} \right)\left[ {x < n} \right]. By well ordering let K be the least to have that property.
    What can be said about K-1?

    If x<0 then do it for -x.
    Would it be that its not in the section? Or am I straying away? I'm looking at my notes and it goes from well ordering to countable without much detail.
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  4. #4
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    If K is the least positive integer having the property that x<K then K-1<K so K-1<x<K.

    You know that there is no integer between K-1~\&~K.
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