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

Sequences

This part of the text covers the elementary theory of sequences:

In 6 Convergence we define sequences and convergence, and see how to prove $\lim s_n=L$ directly from the definition.
In 7 Limit Laws we study the arithemetic of convergent sequences, and prove the limit laws familiar from an introductory calculus course.
In 8 Monotone Convergence we prove the monotone convergence theorem which gives simple conditions that ensure the convergence of a sequence, and use this to study infinite processes and the square root calculating algorithm of the babylonians.
In 9 Subsequences we extend the reach of our theory to cover non-monotone sequences, by decomposing them into subsequences and investigating the resulting limits.
In 10 Cauchy Sequences we define the notion of a Cauchy sequence, and prove it is equivalent to convergence. This lets us study all sorts of new convergent sequences, such as contraction maps.
In 11 \blacklozenge Limits of Limits we get a first look at the complications that arise when multiple limits interact in a single expression. Such limits underlie many interesting situations in analysis, from the theory of power series, to the commutativity of partial derivatives and the ability to differentiate under the integral.