$$$$

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 ?sec-sequences-iterated 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.