I was tasked by the question to prove that the field of rational numbers has the Archimedean property
Proof:
Let $x$ and $y$ $\in$$ Q$ and $n$ be a positive integer. If $Q$ does not have the Archimedean property, then $nx\leq y$
For the case where $nx=y$, then $n+1\gt y$
however as $n+1$ is a positive integer then by contrapositive $Q$ must have the archimedean property
Is this valid and how can I phrase this better?
Best Answer
Perhaps the following is a bit cleaner:
Fix $x,y\in\mathbb{Q}$ with $x>0$. We need to produce a positive integer $n$ with $nx>y$.
By definition of $\mathbb{Q}$, there are integers $a,b,c,d$ with $x=\frac{a}{b}$, $y=\frac{c}{d}$ and $b>0$ and $d>0$.
As $x>0$, we have that $a>0$. Now, there are two cases: