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

1. 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.

2. 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.

3. Originally Posted by SunRiseAir
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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