While I find this book really excellent, there
were some areas where I wanted to give an
emphasis to the material different from that
given in the book; for example, this being
single-variable analysis, I wanted to give
more emphasis to the real line than to abstract
metric spaces. Hence the supplementary material
enclosed here.
The manuscript currently is in a fairly preliminary
stage, and it may never get much beyond it.
Here is a brief table of contents:
Preface.
Interchange of quantifiers.
The Axiom of Completeness.
Supremum and limits.
Upper and lower limit.
Compactness of closed intervals.
The method of interval-halving:
The Heine--Borel Theorem.
The Bolzano-Weierstrass Theorem.
The method of interval-halving:
The Bolzano--Weierstrass Theorem.
Continuous functions.
The Intermediate-Value Theorem.
The method of interval-halving:
the Intermediate-Value Theorem.
The Maximum-Value Theorem.
Uniform continuity, uniform convergence.
Differentiability and continuity.
An Intermediate-Value Theorem for derivatives.
The chain rule for differentiation.
The inverse of a monotone function.
l'Hospital's rule.
The remainder term in Taylor's formula.
The binomial series.
The Riemann integral.
The Riemann integral: an example.
The Newton--Leibniz formula.
Integration by parts.
Compactness product spaces.
Interchanging integration and differentiation.
Bibliography.
List of symbols.
Subject index.
You can find the manuscript
by clicking on the directory link
here;.
this will display a directory listing.
The name of the file
containing the manuscript is analysis.pdf,
and the name of the cover page is cover.pdf;
the file todays_date contains the date these
files were prepared, and you are currently
reading the file pref.html.
You can also download the manuscript directly by
clicking or shift-clicking
here;
its size is 614787 bytes.
(Shift clicking means holding down the shift key while
clicking on the link. It prevents Mozilla from
displaying the file and will initiate a download
instead.)
The manuscript was written in AmS-TeX. Read
here
about how it was turned into PDF (note the subtle
point about dvips).
New York, New York,
Tue Jul 31 19:19:31 EDT 2012
Attila Mate