A Gateway to Higher Mathematics integrates the process of teaching students how to do proofs into the framework of displaying the development of the real number system. The text eases the students into learning how to construct proofs, while preparing students how to cope with the type of proofs encountered in the higher-level courses of abstract algebra, analysis, and number theory. After using this text, the students will not only know how to read and construct proofs, they will understand much about the basic building blocks of mathematics. The text is designed so that the professor can choose the topics to be emphasized, while leaving the remainder as a reference for the students.

Chapter 1. Logic and Techniques For Proofs Chapter 2. Elementary Set Theory Chapter 3. The Development of the Integers Chapter 4. Properties and Applications of Integers Chapter 5. Fields and the Rational Numbers Chapter 6. The Development of the Real Numbers Chapter 7. Some Additional Properties of Real Numbers Appendix A: Proof of the Cantor-Schroder-Bernstein Theorem Appendix B: Using the Axiom of Choice to Prove Some Results About Infinite Sets Appendix C: Completion of the Construction of a Set Obeying the Real Number Postulates

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