$$ \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\FF}{\mathbb{F}} \renewcommand{\epsilon}{\varepsilon} % ALTERNATE VERSIONS % \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}} % \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}} % \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}} % \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}} % \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} \newcommand{\uppersum}[1]{U_{#1}} \newcommand{\lowersum}[1]{L_{#1}} \newcommand{\upperint}[1]{U_{#1}} \newcommand{\lowerint}[1]{L_{#1}} \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} % extra auxiliary and additional topic/proof \newcommand{\extopic}{\bigstar} \newcommand{\auxtopic}{\blacklozenge} \newcommand{\additional}{\oplus} \newcommand{\partitions}[1]{\mathcal{P}_{#1}} \newcommand{\sampleset}[1]{\mathcal{S}_{#1}} \newcommand{\erf}{\operatorname{erf}} $$

analysis

reaching for infinity

Preface

This is the beginnings of a textbook for a 1-year course on real analysis; the current version covers a semester and a half of material. This can be used as a 1-semester course by omitting some of the topics marked in the text as

• \(\extopic\): optional topic independent of the main text (referenced only in later starred sections)
• \(\auxtopic\): content used in main text, but only to prove some supporting or readily-believed fact: these arguments can be skipped or skimmed with little ill effect.
• \(\additional\): additional proof of a result which is proved by different (often cleaner) means elsewhere

The sections ‘Elementary Functions’ present across the second half of the text are self-contained and could be omitted from a course culminating with the Fundamental Theorem of Calculus, but will be an integral part of the eventual year-long course.

If you enrolled in my Spring 2025 course the homework assignments are available here.