Chapter 2: The Real Numbers
The basis for everything we do in analysis is the real number system. In Calculus, most of our understanding of the real numbers is based on geometric intuition, together with simple algebra. In this chapter we shall give axioms for the real numbers (much like for groups or rings) which we can then use to prove useful facts
about them. The ingredients that go into this are the usual arithmetic properties, together with a very subtle property related to the ordering of the reals. This means we have to look at how sets of numbers can be ordered, in careful detail.