Real Analysis

With Proof Strategies

Daniel W. Cunningham

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000294248
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

Typically, undergraduates see real analysis as one of the most difficult courses that a mathematics major is required to take. The main reason for this perception is twofold: Students must comprehend new abstract concepts and learn to deal with these concepts on a level of rigor and proof not previously encountered. A key challenge for an instructor of real analysis is to find a way to bridge the gap between a student's preparation and the mathematical skills that are required to be successful in such a course.

Real Analysis: With Proof Strategies provides a resolution to the "bridging-the-gap problem." The book not only presents the fundamental theorems of real analysis, but also shows the reader how to compose and produce the proofs of these theorems. The detail, rigor, and proof strategies offered in this textbook will be appreciated by all readers.

Features

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• Explicitly shows the reader how to produce and compose the proofs of the basic theorems in real analysis

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• Suitable for junior or senior undergraduates majoring in mathematics.

CHAPTER 1

Proofs, Sets, Functions, and Induction

In this introductory chapter, we review and identify the preliminaries that are essential for real analysis. In particular, we review inequalities, sets, functions, and proof by mathematical induction. These preliminary topics should be familiar, as should be the basic proof techniques that are used in mathematics. A review of logic and proof is presented in Appendix C on page 253. The most important proof strategies that are applied in this text can be found in the appendix starting on page 256.

1.1 Proofs

1.1.1 Important Sets in Mathematics

The set concept is frequently used in mathematics. A set is a well-defined collection of objects. The items in such a collection are called the elements or members of the set. The symbol “∈” is used to indicate membership in a set. Thus, if A is a set, we write x∈A to declare that x is an element of A. Moreover, we write x∉A to assert that x is not an element of A. In mathematics, a set is typically a collection of mathematical objects, for example, numbers, functions, or other sets. Certain se...