I learned calculus from the excellent Calculus [13] of D. R. Dickinson and its inspiring author. My first glimpse of analysis was in Hardy's Pure Mathematics [24] read when I was too young to really understand it. I learned elementary analysis from Ferrar's A Textbook of Convergence [18] (an excellent book for those making the transition from school to university, now, unfortunately, out of print) and Burkill's A First Course in Mathematical Analysis [10].
The books of Kolmogorov and Fomin [31] and, particularly, Dieudonne [14] showed me that analysis is not a collection of theorems but a single coherent theory. Stromberg's book An Introduction to Classical Real
Analysis [46] 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 popular1.