Rudin gives the definition of a Dedekind Cut to be:
A set of rational numbers is said to be a cut if
(I) $\alpha$ contains at least one rational, but not every rational;
(II) if $p\in\alpha$ and $q<p$ (q rational), then $q\in\alpha$;
(III) $\alpha$ contains no largest rational.
I'm confused as to how a set of rationals can contain no largest rational, yet not contain every rational.