Why Don’t We Follow the Order in the Textbook?

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Why Don’t We Follow the Order in the Textbook?
You may be wondering why we are not following the exact order in the textbook. For that matter, you may
be wondering why we are not starting the class with the textbook. I will answer the second question first:
The text, Elementary Analysis, by Kenneth Ross, does not cover some introductory material I feel is helpful
for this course. I will be taking this material from Steven Lay’s book, Analysis with an Introduction to Proof.
Now for the first question. There are several reasons for ordering the topics as I have. The primary reason
is that most of you took the calculus sequence here at the University of Utah, using the text by Rigdon and
Purcell. That being the case, I have chosen to present the material for this course in the same order as it
was presented in the calculus classes. In particular, you may notice that we will cover section 20 (Limits of
Functions) before covering section 17 (Continuity). You may recall that in calculus, we defined a function
f to be continuous at a point a if lim f (x) = f (a). This is the definition you are used to, and it seems much
x→a
more natural than the one in the book. Therefore, it makes sense to cover these topics in the order I have
chosen.
You may also notice that we are postponing our coverage of series and power series (sections 14-15, and
23-25) until the end of the course. The brevity of this term prevents us from covering all topics of the course
in the homework, if not the lectures. Therefore, I have chosen to delay our coverage of series and power
series knowing full-well that these topics will not be covered in the homework. In so doing, we will definitely
be able to cover the Derivative, the Integral, and the Fundamental Theorem of Calculus, the three essential
topics in calculus.