Rudin's Construction of Real Numbers At the end of chapter 1 of Principles of Mathematical Analysis, Rudin provides a proof of the construction of real numbers. The first step in the proof is to define members of $\mathbb{R}$ that are subsets of $\mathbb {Q}$ known as cuts. Rudin gives the following definition of a cut:


*

*$\alpha$ is not empty, and $\alpha \neq \mathbb{Q}$.

*If $p \in \alpha, q \in \mathbb {Q}$, and $q < p$, then $ q \in \alpha$

*If $p \in \alpha$, then $p < r$ for some $r \in \alpha$.


The letters $p, q, r, \ldots$ will always denote rational numbers, and $\alpha, \beta, \gamma, \ldots$ will denote cuts.
My question is the following:
Doesn't the definition of the cut provided lead to the conclusion that each cut contains all rational numbers? 
In other words, that $\alpha = \mathbb {Q}$ for all $\alpha$?
The third property of cuts provided shows that there is no maximal element in a cut. The second property implies that given an element in a cut, all rational numbers less than that element are in the cut. Doesn't this clearly imply that cuts include all the rationals?
I was hoping someone could clear this up. Thanks.

No, of course not. For example, the set of negative rational numbers are a cut.
