## A Gateway to Higher MathematicsA 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. |

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### Contents

Logic and Techniques for Proofs | 2 |

Elementary Set Theory | 21 |

The Development of the Integers | 71 |

Properties and Application of Integers | 139 |

Fields and the Rational Numbers | 197 |

The Development of the Real Numbers | 223 |

Some Additional Properties | 259 |

Proof of the CantorSchroderBernstein | 287 |

305 | |

### Common terms and phrases

asked to show Axiom of Choice base b series bijection binary operation card(A Cauchy sequence commutative complete ordered field complete the proof concept consider convergent countable set course Define the set Definition demonstrate denote dn+i equation equivalence class equivalence relation example exists express G A7 Hence inductive step Inequality infinite sets integer space intuitive inverse isomorphism least element Lemma linear order lower bound mathematical induction n G N nonempty set notation Note one-to-one ordered integral domain ordered pair Peano space positive integer proof of Theorem Property Prove Theorem rational numbers rational space reach a contradiction reader is asked real number system real sequence Recall Recursion Theorem result holds ring Section sequence an}~=i set theory shown statement student subring Suppose to reach tion truth table unique write