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Math Help - Can anyone help me with this proof?

  1. #1
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    Question Can anyone help me with this proof?

    Give a complete, formal proof of the following statement: If
    a is an element of the real numbers, then there exists a negative

    integer
    n such that n < a.

    I know its related to the archimedean property but cant find the proof anywhere.

    Cheers!
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  2. #2
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    Quote Originally Posted by Nappy View Post
    Give a complete, formal proof of the following statement: If
    a is an element of the real numbers, then there exists a negative

    integer
    n such that n < a.

    I know its related to the archimedean property but cant find the proof anywhere.

    Cheers!
    Maybe my question is dumb: Why can't we just take n=\lceil a\rceil-1?
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  3. #3
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    Quote Originally Posted by Nappy View Post
    Give a complete, formal proof of the following statement: If a is an element of the real numbers, then there exists a negative integer n such that n < a. I know its related to the archimedean property but cant find the proof anywhere.
    Clearly if a\ge 0 then -1<a works.
    So suppose that a<0 then 0<-a.
    Using the Archimedean property, we know \left( {\exists N \in \mathbb{Z}^ +  } \right)\left[ { - a < N} \right] .
    You finish.
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