Chapter 2
Sequences
Sections 2.1- 2.7, through Theorem 2.7.4, contain essential
core material. Indeed, the concepts and methodology intro-
duced here lie at the very heart of the subject, and will be
used throughout the remainder of the book. This chapter also
contains some material labeled "*" that may be covered in
courses lasting more than one semester. Section 2.8 should
be learned sometime in a student's career, but not necessar-
ily here. Upper and lower limits of sequences are covered in
(optional) Section 2. 9.In this chapter we experience our first encounter with the notion of "limit."
This notion is a central concept in the subject of analysis. It will appear in
several guises throughout the course. I believe, as do many mathematicians,
that the concept and techniques of limits are most easily learned in the context
of (infinite) sequences. Later forms and properties of limits are then readily
seen as extensions of those learned in this context.
We shall develop the theory of sequences to a level where it can be seen
as a powerful tool in analysis. Much of the power of analysis is anticipated in
the theory of sequences, and many of its deepest results can be formulated in
the language and conceptual framework of sequences. In fact, the concepts and
methods presented in this chapter will be used in every remaining chapter of
the book.
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