Density of Rationals in Reals Proof

Question

Use Archimedean Property property to prove $Q$ is dense in $R$.

Proof

We want to show that there exists a $m\in Z$ and a $n\in N$ such that $a<\frac{m}{n}<b$

1. Assume $x,y\in R$ and $x<y$
2. Choose $n$ such that $n>\frac{1}{y-x}$ and $n\in N$
   1. Then $\frac{1}{n}<y-x$
   2. Then $x<y-\frac{1}{n}$
3. Choose $m$ such that $m-1\le nx<m$
   1. Then $(m-1)\le nx<m$
   2. Then $x<\frac{m}{n}$
4. Consider $m-1<n(x)$
   1. Then $m<n(x)+1$
   2. Then $m<n(y-\frac{1}{n})+1$
   3. Then $m<ny-1+1$
   4. Then $m<ny$
   5. Then $\frac{m}{n}<y$
5. Thus $x<\frac{m}{n}<y$
6. $\frac{m}{n}\in Q$ $\square$