to earlier versions of this book. My e-mail address is twk@dpmms. cam. ac .uk
and I shall try to keep a list of corrections accessible from an AMS page
at www.ams.org/bookpages/gsm-62 as well as from my home page page at
I learned calculus from the excellent Calculus  of D. R. Dickinson and
its inspiring author. My first glimpse of analysis was in Hardy's Pure Math-
ematics  read when I was too young to really understand it. I learned
elementary analysis from Ferrar's A Textbook of Convergence  (an ex-
cellent book for those making the transition from school to university, now,
unfortunately, out of print) and Burkill's A First Course in Mathematical
Analysis . The books of Kolmogorov and Fomin  and, particularly,
Dieudonne  showed me that analysis is not a collection of theorems but a
single coherent theory. Stromberg's book An Introduction to Classical Real
Analysis  lies permanently on my desk for browsing. The expert will
easily be able to trace the influence of these books on the pages that follow.
If, in turn, my book gives any student half the pleasure that the ones just
cited gave me, I will feel well repaid.
Cauchy began the journey that led to the modern analysis course in
his lectures at the Ecole Polytechnique in the 1820's. The times were not
propitious. A reactionary government was determined to keep close control
over the school. The faculty was divided along fault lines of politics, religion
and age whilst physicists, engineers and mathematicians fought over the
contents of the courses. The student body arrived insufficiently prepared
and then divided its time between radical politics and worrying about the
job market (grim for both staff and students). Cauchy's course was not
Everybody can sympathise with Cauchy's students who just wanted to
pass their exams and with his colleagues who just wanted the standard
material taught in the standard way. Most people neither need nor want to
know about rigorous analysis. But there remains a small group for whom the
ideas and methods of rigorous analysis represent one of the most splendid
triumphs of the human intellect. We echo Cauchy's defiant preface to his
printed lecture notes.
As to the methods [used here], I have sought to endow them
with all the rigour that is required in geometry and in such
a way that I have not had to have recourse to formal ma-
nipulations. Such arguments, although commonly accepted
... cannot be considered, it seems to me, as anything other
than [suggestive] to be used sometimes in guessing the truth.
Belhoste's splendid biography  gives the fascinating details.