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.