density of the rationals on the reals proof

Solution 1:

The meaning of $x$ is different between the two versions of the Archimedean property you're quoting. I will switch to different notation so that you can see the similarity.

The second property says:

For all $r \in \mathbb R$, there is an $n \in \mathbb N$ such that $n > r$. Every real number has an integer bigger than it.

The first property says:

For all $r \in \mathbb R$, and for all $u \in \mathbb R$ with $u > 0$, there is an $n \in \mathbb N$ such that $nu > r$. Every real number has an integer multiple of $u$ bigger than it.

The second property is a special case of the first, with $u=1$. (Of course, you can also use the second property to prove the first, by applying it to $\frac ru$.)

The intuition behind the Archimedean property is this: all real numbers live in the same world. If you pick any real number as your unit, you will be able to measure all other real numbers in those units.

But we want our unit $u$ to be a positive real number, because we want $nu$ to get arbitrarily large as $n \in \mathbb N$ increases. This would not be necessary if we took $n \in \mathbb Z$ instead.