I just read about the construction of the real numbers in Enderton's Elements of Set Theory, and am now trying to go through all the exercises. Enderton chooses to construct the reals with left sides of Dedekind cuts. His definition is $y$ is a Dedekind cut if $\emptyset\neq y\neq\mathbb{Q}$, $y$ has no largest member, and $y$ is downward closed, i.e., $q\in y\wedge r. The following, question 19 of chapter 5 has me a bit baffled.
Assume that $p$ is a positive rational number. Show that for any real number $x$ there is some rational $q$ in $x$ such that $p+q\not\in x$.
I have this intuitive argument in my head. I guess I want to take $q\in x$, preferably closer to the right in $x$ on the rational line than not, and add it to $p$. If $p+q\not\in x$, then I am done. Otherwise, since $x$ has no greatest element, I can find q'\in x such that q'>q, and add it to $p$ until I finally get a $q$ that pushes $p+q$ out of $x$, supposing of course that this process eventually ends. But this is horribly informal, and I have very little experience with analysis, and can't see a way to formalize it (if this is even the correct approach.)
Could someone please explain how to make this argument more rigorous, or perhaps point out a better way? Thanks.