The approach Rudin takes is this.
Describe the field axioms and the ordered field axioms. Claims that there exists an ordered field which satisfies the least upper bound property. This field is called the real numbers and is denoted as R (note this is well-defined because any ordered field satisfying the LUB property is isomorphic to R). Rudin does not construct R in chapter 1. He leaves that to an appendix by using Dedekind cuts. Note there is also an exercise in Chapter 3 which gives an alternative construction, and in fact, a generalizable construction; this one is known as the equivalence class of Cauchy sequence construction (full and glorious details may be found, for example, in Introductory Real Analysis by Kolmogorov)