Dedekind Cuts in Rudin' analysis - Step 4 This question refers to the construction of $\mathbb{R}$ from $\mathbb{Q}$ using Dedekind cuts, as presented in Rudin's "Principles of Mathematical Analysis" pp. 17-21.
More specifically, in the last paragraph of step 4, Rudin says that for $\alpha$ a fixed cut, and given $v \in 0^*$, setting $w=- v / 2$, there exists an integer $n$ such that $nw \in \alpha$ but $(n+1)w$ is not inside $\alpha$. Rudin says that this depends on the Archimedean property of the rationals, however he has not proved it. 
Could somebody prove the existence of the integer $n$?

If $v\in 0^*$, then $v<0$, so $w>0$. Let $\gamma=\sup \alpha$.  The Archimedean property says that for any $\gamma$ there is some integer $m$ such that $mw\geq\gamma$. Since $\mathbb{N}$ is well-ordered, there is a smallest $m$ such that $mw\geq\gamma$, call this $n+1$. This means that $nw<\gamma$, and hence $nw\in \alpha$. But also $(n+1)w\notin \alpha$ since it is larger than $\gamma$ (OK, so there might be some subtleties with endpoints you should think about, but essentially this is the idea).
