The Rationals Are Dense in the Reals

Suppose we take two real numbers $x$ and $y$, and let $x>y$ without any loss of generality. Then we have some natural number $n$ such that $n(x-y)>1$. (For example if $x=3.1$ and $y=2.8$, then one candidate is $n=4$.) Now by the Archimedean property, there exists two natural numbers $n_1$ and $n_2$ such that $-n_2<ny<n_1$. Since $-n_2<0$, there must exists some integer $N$ such that $-n_2\leq N\leq n_1$ and $N-1\leq ny< N$ and hence we have $ny<N<nx$, now dividing both by $n$ we get $x<N/n<y$. Thus, we now have a rational number between two real numbers, so the rationals are dense in the reals.