New York University: Lawrence Tsang's "Real Numbers"

Read pages 1-7 of the text. The first 6 pages discuss the field and order axioms for real numbers. The Completeness Axiom on page 6 is what distinguishes the rational numbers from the real numbers - the latter is COMPLETE, while the former is not.

Click http://web.archive.org/web/20150513235107/http://math.nyu.edu/~tsang/classes/precalc1/session1.pdf link to open resource.