Background I was an eighteen-year-old freshman when I began studying analysis. I had arrived at Columbia University ready to major in physics or perhaps engineering. But my seduction into mathematics began immediately with Lipman Bers’ calculus course, which stood supreme in a year of exciting classes. Then after the course was over, Professor Bers called me into his o?ce and handed me a small blue book called Principles of Mathematical Analysis by W. Rudin. He told me that if I could read this book over the summer,understandmostofit,andproveitbydoingmostoftheproblems, then I might have a career as a mathematician. So began twenty years of struggle to master the ideas in “Little Rudin. ” I began because of a challenge to my ego but this shallow reason was quickly forgotten as I learned about the beauty and the power of analysis that summer. Anyone who recalls taking a “serious” mathematics course for the ?rst time will empathize with my feelings about this new world into which I fell. In school, I restlessly wandered through complex analysis, analyticnumbertheory,andpartialdi?erentialequations,beforeeventually settling in numerical analysis. But underlying all of this indecision was an ever-present and ever-growing appreciation of analysis. An appreciation thatstillsustainsmyintellectevenintheoftencynicalworldofthemodern academic professional. But developing this appreciation did not come easy to me, and the p- sentation in this book is motivated by my struggles to understand the viii Preface most basic concepts of analysis. To paraphrase J.

Table of Contents:
Numbers and Functions, Sequences and Limits.- Mathematical Modeling.- Natural Numbers Just Aren’t Enough.- Infinity and Mathematical Induction.- Rational Numbers.- Functions.- Polynomials.- Functions, Functions, and More Functions.- Lipschitz Continuity.- Sequences and Limits.- Solving the Muddy Yard Model.- Real Numbers.- Functions of Real Numbers.- The Bisection Algorithm.- Inverse Functions.- Fixed Points and Contraction Maps.- Differential and Integral Calculus.- The Linearization of a Function at a Point.- Analyzing the Behavior of a Population Model.- Interpretations of the Derivative.- Differentiability on Intervals.- Useful Properties of the Derivative.- The Mean Value Theorem.- Derivatives of Inverse Functions.- Modeling with Differential Equations.- Antidifferentiation.- Integration.- Properties of the Integral.- Applications of the Integral.- Rocket Propulsion and the Logarithm.- Constant Relative Rate of Change and the Exponential.- A Mass-Spring System and the Trigonometric Functions.- Fixed Point Iteration and Newton’s Method.- Calculus Quagmires.- You Want Analysis? We’ve Got Your Analysis Right Here.- Notions of Continuity and Differentiability.- Sequences of Functions.- Relaxing Integration.- Delicate Limits and Gross Behavior.- The Weierstrass Approximation Theorem.- The Taylor Polynomial.- Polynomial Interpolation.- Nonlinear Differential Equations.- The Picard Iteration.- The Forward Euler Method.