Why does every course at the university teach the construction of the real numbers as equivalence classes of cauchy sequences of rational numbers? Why no dedekind cuts?

They're bags more intuitive and easier to manipulate....

I know the cauchy sequences generalize better, but I was convinced up until about a year ago that there was something wrong with cuts.

Is there an operation they can't handle? Does anyone know of any constructions (in the intuitionist sense) other than these?....

Answers on a postcard please.

This is pretty interesting, though I find most of the other constructions bring little compared to the Cauchy series one (the surreal numbers thing is pretty cool though). I really don't see any problem with Dedekind cuts apart from the fact that, as you said, the property which we want to be "obvious" (the upper bound one) isn't our general definition of completeness (as is the case with Cauchy sequences).

I thought that they did mention Dedekind cuts somewhere, but that may have been a dream. However, the whole "existence of suprema for subsets of R bounded above" is basicly logically equivalent. Or was that a dream too? I really suck at this degree course. It requires you to be awake at particular times, and to be vaguely in touch with reality. I think I'm learning a lot of maths here, which is great, but I feel like I'm a radian or two out of phase with the course.

I like to prove it prove it.

Yeah, it's equivalent, and pretty easy to prove (I think it was in one of the Analysis I workbooks for that matter). The idea is that if you construct the reals as cuts then this property is obvious from the construction and you need to prove completeness in terms of Cauchy sequences and vice versa, so it doesn't matter that much in the end (I still like the Cauchy sequences one a little bit better :P).