Download Full Text (977 KB)

Download Note to Students (41 KB)

Download Preliminaries (77 KB)

Download Direct Proofs (102 KB)

Download Using Logical Equivalencies in Proofs (88 KB)

Download Proof by Contradiction (90 KB)

Download Using Cases in Proofs (79 KB)

Download Mathematical Induction (113 KB)

Download Injective and Surjective Functions (977 KB)

Download Appendix A: Guidelines for Writing Mathematical Proofs (75 KB)

Download Appendix B: Answers and Hints for the Practice Problems (173 KB)

Download Index (35 KB)


This little book is not intended to be a textbook for a course dealing with an introduction to constructing and writing mathematical proofs. It is intended to be a reference book for students who need to construct and write proofs in their upper division mathematics courses. So it is assumed that students who use this as a reference have already taken an “introduction to proofs” course.

With the exception of Chapter 1, each chapter in the book has a description of a proof technique along with some justification as to why it is a valid proof method. There are then one or two completed proofs written according to the writing guidelines for mathematical proofs in Appendix A. The intent is to illustrate a well-written proof for that particular proof method. Each chapter then ends with three to five practice problems, most of which deal with mathematical proofs. Completed proofs (or solutions) for the practice problems are contained in Appendix B. So, students can check their work or see other examples of well-written proofs. Chapter 1 contains most of the definitions used in the first six chapters of this book and a short summary of some logic that is pertinent to constructing mathematical proofs.

The proofs in this book primarily use the concepts of even and odd integers, the concept of one integer dividing another, and the concept of congruence in the integers. Most of this book is based on material in chapter 3 of the book Mathematical Reasoning: Writing and Proof, Version 2.1 by Ted Sundstrom, which is a textbook for an “introduction to proofs” course. It is free to download as a pdf file at

A printed version of this book is also available on for $22 at

Finally, there is a website for Mathematical Reasoning: Writing and Proof, Version 2.1. Please visit and click on the TEXTBOOKS button in the upper right corner. This website contains useful resources for an introduction to mathematical proofs course, and some of these resources could be useful for students in upper division mathematics courses.

Publication Date



Grand Valley State University Libraries


Allendale, MI

Creative Commons License

Creative Commons Attribution-Noncommercial-Share Alike 3.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.


proofs, mathematical proofs



Constructing and Writing Mathematical Proofs: A Guide for Mathematics Students

Included in

Mathematics Commons