Since the subject of definitions arose, I am curious about how and what users of this list tell their students about the real numbers. I have been teaching real analysis a few times in recent years, and spend a lot of time on the reals, since constructing them lays an excellent groundwork for the whole concept of delta-epsilon stuff. I use an approach developed by Gabe Stolzenberg which defines reals as families of intervals of rational numbers. It combines, it seems to me, the best features of both the Dedekind and Cauchy approaches. I'd very much like to hear what others are doing.