TITLE: A question on Dedekind's cuts
QUESTION [0 upvotes]: Rudin write that the members of $\mathbb R$ will be certain subsets of $\mathbb Q$, called cuts. 
A cut is, by definition, any set $\alpha\subset \mathbb Q$ with the following three properties.
(I) $\alpha$ is not empty, and $\alpha\neq \mathbb Q$
(II) If $p\in \alpha$ and $q<p$ then $q\in\alpha$
(III) If $p\in \alpha$ then $p<r$ for some $r\in \alpha$
If $\alpha\in \mathbb R^{+}$ and $\beta \in \mathbb R^{+}$ we define $\alpha\beta$ to be the set of all $p$ such that $p\leqslant rs$ for some choices of $r\in \alpha$, $s\in \beta$, $r>0$,$s>0$.
Affirms that for any $\alpha\in \mathbb R^{+}$ and $\alpha\neq 0^{*}$ there is $\beta\in \mathbb R^{+}$ such that $\alpha\beta=1^{*}$.
How to define this cut $\beta$?

REPLY [1 votes]: Answer.
$$
\beta=\big\{p\in\mathbb Q : pq\le 1\,\,\text{for all}\,\,q\in\alpha\cap\mathbb Q^+\big\}.
$$
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