Properties of the Real Numbers
In order to prove the theorems of real analysis we need to make use of the properties of the real numbers, and hence a rigorous treatment of real analysis requires a rigorous foundation for the real numbers. As discussed in Section 1.1, we offer in this text three ways to enter into the study of the real numbers. The first two ways were given in Chapter 1, namely, Entry 1 in Section 1.2, which begins with an axiomatic treatment of the natural numbers, and Entry 2 in Section 1.4, which begins with an axiomatic treatment of the integers. Starting with either Entry 1 or Entry 2, the culmination of Chapter 1 is the construction of the real numbers, together with the proofs of the core properties of these numbers, stated in Theorem 1.7.6 and Theorem 1.7.9.
KeywordsReal Number Natural Number Rational Number Algebraic Property Real Analysis
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