John A Rock

Arbitrarily Close: An Introduction to Real Analysis, 2025. Download the PDF, if you like! 

Hardcover and paperback versions are available for purchase from 619 Wreath.

Also, this collection of Desmos and GeoGebra activities supplement the textbook.

License

Arbitrarily Close: An Introduction to Real Analysis is licensed under a Creative Commons Attribution 4.0 International License.

Synopsis

Arbitrarily Close: An Introduction to Real Analysis is for readers who prefer thorough explanations and an abundance of detail beyond what the current canon on analysis generally offers. Concepts are built from a formal definition for “arbitrarily close” to parse introductions to the core and notoriously difficult concepts of convergence, compactness, continuity, limits, derivatives, integrals, uniform convergence, and series.

The structure follows a definition-example/theorem/lemma-scratch work-proof pattern. In particular, scratch work is used to motivate, comment upon, and provide preliminary work for proofs. Included are approximately 150 figures and 20 QR codes with links to dynamic Desmos and GeoGebra activities, all designed to help the reader find their own intuition for real analysis. Color is used in figures and to differentiate definitions, examples, theorems, remarks, and scratch work in order to aid with organization and the ability to reference material quickly.

From the reviewer

"Arbitrarily Close is a gentle introduction to undergraduate real analysis... The introduction of numerous interactive Desmos and GeoGebra applets give concrete and intuitive realizations of abstract arguments and results. Many of the arguments and proofs throughout this text were authored by former students, further highlighting the student-centered focus. In every way, Arbitrarily Close's guiding philosophy of student accessibility and understanding is readily apparent..."