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Math Help - Prove that the set of rational numbers that are not integers is dense in the reals

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    Prove that the set of rational numbers that are not integers is dense in the reals

    This is from Fitzpatrick's Advanced Calculus, where it has already been shown that the rationals are dense in \mathbb{R}:

    Prove that the set \mathbb{Q}\backslash\mathbb{Z} of rational numbers that are not integers is dense in \mathbb{R}.

    So far I have:


    We know the rationals \mathbb{Q} are dense in \mathbb{R}. In the case where there are no integers in the set (a,b) where a<b, then (a,b)\backslash\mathbb{N}=(a,b) and so \mathbb{Q}\cap (a,b)\not=0 as shown previously.

    In the case where there is at least one integer m in the set (a,b), we want to show that there is also a non-integer in (a,b).


    I am not sure where to go from here, or if dividing it into cases is the correct approach. Could anyone give me some pointers? Thanks!
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    Re: Prove that the set of rational numbers that are not integers is dense in the real

    Quote Originally Posted by Ragnarok View Post
    This is from Fitzpatrick's Advanced Calculus, where it has already been shown that the rationals are dense in \mathbb{R}:

    Prove that the set \mathbb{Q}\backslash\mathbb{Z} of rational numbers that are not integers is dense in \mathbb{R}.

    Suppose that t\in \mathbb{Q}\setminus\mathbb{Z}. Then using the floor function t  \notin \mathbb{Z} and so \left\lfloor t \right\rfloor  < t < \left\lfloor t \right\rfloor  + 1

    If \delta=\min\{t-\left\lfloor t \right\rfloor,\left\lfloor t \right\rfloor+1-t\} then t\in O=\left( {t - \delta ,t + \delta } \right)\cap\mathbb{Z}=\emptyset.

    Now use the density of \mathbb{Q} in \mathbb{R} with respect to O.
    Last edited by Plato; May 18th 2013 at 03:03 PM.
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    Re: Prove that the set of rational numbers that are not integers is dense in the real

    Quote Originally Posted by Ragnarok View Post
    This is from Fitzpatrick's Advanced Calculus, where it has already been shown that the rationals are dense in \mathbb{R}:

    Prove that the set \mathbb{Q}\backslash\mathbb{Z} of rational numbers that are not integers is dense in \mathbb{R}.

    To do it your way, this is an edit.
    If (a,b)\cap\mathbb{Z}=\emptyset then you are done.

    If (a,b)\cap\mathbb{Z}\ne\emptyset then let j be the least integer in (a,b)\cap\mathbb{Z}.

    Then If (a,j)\cap\mathbb{Z}=\emptyset and (a,j)\subset(a,b)
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    Re: Prove that the set of rational numbers that are not integers is dense in the real

    Thank you! I think I got it.
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