This book is a rigorous introduction to real analysis, suitable for a onesemester course at the second-year undergraduate level, based on my experience of teaching this material many times in Australia and Canada. My aim is to give a treatment that is brisk and concise, but also reasonably complete and as rigorous as is practicable, starting from the axioms for a complete ordered field and a little set theory.

Along with epsilons and deltas, I emphasise the alternative language of neighbourhoods, which is geometric and intuitive and provides an introduction to topological ideas. I have included a proper treatment of the trigonometric functions. They are sophisticated objects, not to be taken for granted. This topic is an instructive application of the theory of power series and other earlier parts of the book. Also, it involves the concept of a group, which most students won't have seen in the context of analysis before.

There may be some novelty in the gentle, example-based introduction to metric spaces at the end of the book, emphasising how straightforward the generalisation of many fundamental notions from the real line to metric spaces really is. The goal is to develop just enough metric space theory to be able to prove Picard's theorem, showing how a detour through some abstract territory can contribute back to analysis on the real line.

Needless to say, I claim no originality whatsoever for the material in this book.