Problems and Solutions for Undergraduate Real Analysis I

The aim of Problems and Solutions for Undergraduate Real Analysis I, as the name reveals, is to assist undergraduate students or first-year students who study mathematics in learning their first rigorous real analysis course. The wide variety of problems, which are of varying difficulty, include the following topics:

• Elementary Set Algebra
• The Real Number System
• Countable and Uncountable Sets
• Elementary Topology on Metric Spaces
• Sequences in Metric Spaces
• Series of Numbers
• Limits and Continuity of Functions
• Differentiation
• the Riemann-Stieltjes Integral

Furthermore, the main features of this book are listed as follows:

1. The book contains 230 problems, which cover the topics mentioned above, with detailed and complete solutions. As a matter of fact, my solutions show every detail, every step and every theorem that I applied.
2. Each chapter starts with a brief and concise note of introducing the notations, terminologies, basic mathematical concepts or important/famous/frequently used theorems (without proofs) relevant to the topic.
3. Three levels of difficulty have been assigned to problems so that you can sharpen your mathematics step-by-step.
4. Different colors are used frequently in order to highlight or explain problems, examples, remarks, main points/formulas involved, or show the steps of manipulation in some complicated proofs. (ebook only)
5. An appendix about mathematical logic is included. It tells students what concepts of logic (e.g. techniques of proofs) are necessary in advanced mathematics.