At the end of the introduction, I give my reasons for choosing the Dedekind cut definition of the real line over the other two classical definitions. Here I will emphasize the reasoning behind how I present Dedekind cuts. One of the things that I have noticed in teaching students is that they sometimes disbelieve the statement that .999...=1, which is not particularly surprising when one considers that many studies on high school students show that this equivalence is very hard to get people to accept and understand.
One of the fundamental ideas to get across in this section (and the class) is that decimal representations of real numbers are not necessarily unique. Moreover, a goal is to have students answering questions that a lack of this understanding affects. For example, we discussed in class the problem of adding .777777... and .8888.... and how (middle and high school) students will sometimes answer that the sum of these two is 1.66....5. As sometimes happens, this term was particularly successful because the day after having this discussion in class, one of the students in my class saw this example carried out by a student in the secondary class she was observing. Unfortunately, it became clear to me as the semester wore on that many of the cooperating teachers for my students didn't believe that .999...=1 (at least based on student reports - I should probably note that most teachers would report (and probably teach) that .999...=1, at least to mathematicians). Again, this emphasizes the importance of dealing with this issue in the capstone class.
The other idea that I try to get across in the section on Dedekind cuts is the importance of approximating real numbers and how such approximations can change answers. In particular, I discuss the case of the faulty Pentium computer chip in the 1990s, and I also discuss how these things affect calculators (as you can see on the homework). At least in my classes, a lack of understanding of the difficulties that approximations cause is something I have seen consistently, although curiously, at Bowling Green, I find that students are too ready to believe that their calculators are correct, while at MSU, I think they are too quick to assume that round off error is a problem.