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Math Help - Is there a real number r ⊂R+, that is smaller than all positive rational numbers?

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    Is there a real number r ⊂R+, that is smaller than all positive rational numbers?

    I was wondering if there exists a real number greater than zero that is smaller than every positive rational number. My guess is that there is, because:


    Consider the rational number 1/n, such that n ∈N (whatever n happens to be).

    Now, if you divide that number by Euler's number, you get an irrational number, right? And isn't 1/(en) always smaller than 1/n, since e > 0?

    The reason why I would doubt this is that there is no largest natural number, which means there is no smallest rational number. Still, for any
    n ∈N, there is a number 1/(en) that is smaller than 1/n, but on the other hand, both approach zero as n approaches infinity, so I'm not sure on this.


    Thanks for any input.
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    Between all rational numbers are irrational numbers. So that means that if there was a smallest possible positive rational number, there would have to be irrational numbers between it and 0.
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    Quote Originally Posted by SunRiseAir View Post
    I was wondering if there exists a real number greater than zero that is smaller than every positive rational number. My guess is that there is, because:


    Consider the rational number 1/n, such that n ∈N (whatever n happens to be).

    Now, if you divide that number by Euler's number, you get an irrational number, right? And isn't 1/(en) always smaller than 1/n, since e > 0?
    but then again, there exists a positive rational number smaller than 1/(en) .
    The reason why I would doubt this is that there is no largest natural number, which means there is no smallest rational number. Still, for any
    n ∈N, there is a number 1/(en) that is smaller than 1/n, but on the other hand, both approach zero as n approaches infinity, so I'm not sure on this.


    Thanks for any input.
    ...
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