We want you to reason with mathematics. We are not trying to get everyone to give
formalized proofs in the sense of contemporary mathematics; ‘proof’ in this course means
‘convincing argument.’ We expect you to use correct reasoning and to give careful explanations.
The projects bring out these issues in the way we find best for most students,
but the pure mathematical questions also interest some students. This book of mathematical
“background” shows how to fill in the mathematical details of the main topics from
the course. These proofs are completely rigorous in the sense of modern mathematics –
technically bulletproof. We wrote this book of foundations in part to provide a convenient
reference for a student who might like to see the “theorem - proof” approach to calculus.