A Note to the Instructor

In this book, I have assembled a collection of what I find to be compelling mathematical

statements with interesting elementary proofs, illustrating diverse proof methods and in-

tended to develop a beginner’s proof-writing skills. All who aspire toward mathematics,

who want to engage fully with the mathematical craft by undertaking a mathematical anal-

ysis and constructing their own proofs of mathematical statements, will benefit from this

text, whether they read it as part of a university proof-writing course or study it on their

own.

I should like to emphasize, however, that the book is not an axiomatic development of its

topics from first principles. The reason is that, while axiomatic developments certainly in-

volve proof writing, I find that they are also often burdened, especially in their beginnings,

with various tedious matters. Think of the need, for example, to establish the associativ-

ity of integer addition from its definition. I find it sensible, in contrast, to separate the

proof-writing craft in its initial or introductory stages from the idea that an entire mathe-

matical subject can be developed from weak axiomatic principles. I also find it important

to teach proof writing with mathematically compelling, enjoyable examples, which can in-

spire a deeper interest in and curiosity about mathematics; students will then be motivated

to work through other examples on their own.

So the proofs in this book are not built upon any explicitly given list of axioms but, rather,

appeal to very general mathematical principles with which I expect the reader is likely

familiar. My hope is that students, armed with the proof-writing skills they have gained

from this text, will go on to undertake axiomatic treatments of mathematical subjects, such

as number theory, algebra, set theory, topology, and analysis.

The book is organized around mathematical themes, rather than around methods of proof,

such as proofs by contradiction, proofs by cases, proofs of if-then statements, or proofs of

biconditionals. To my way of thinking, mathematical ideas are best conceived of and

organized mathematically; other organizational plans would ultimately be found artificial.

I do not find proofs by contradiction, for example, to be a natural or robust mathematical

category. Such a proof, after all, might contain essentially the same mathematical insights