Introduction to Analysis
American Mathematical Soc., 2009 - 240 pages
Introduction to Analysis is designed to bridge the gap between the intuitive calculus usually offered at the undergraduate level and the sophisticated analysis courses the student encounters at the graduate level. In this book the student is given the vocabulary and facts necessary for further study in analysis. The course for which it is designed is usually offered at the junior level, and it is assumed that the student has little or no previous experience with proofs in analysis. A considerable amount of time is spent motivating the theorems and proofs and developing the reader's intuition. Of course, that intuition must be tempered with the realization that rigorous proofs are required for theorems. The topics are quite standard: convergence of sequences, limits of functions, continuity, differentiation, the Riemann integral, infinite series, power series, and convergence of sequences of functions. Many examples are given to illustrate the theory, and exercises at the end of each chapter are keyed to each section. Also, at the end of each section, one finds several Projects. The purpose of a Project is to give the reader a substantial mathematical problem and the necessary guidance to solve that problem. A Project is distinguished from an exercise in that the solution of a Project is a multi-step process requiring assistance for the beginner student.
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1–1 function absolutely convergent accumulation point alternating series test Bolzano–Weierstrass Theorem calculus Cauchy sequence Chapter compact continuous at xo continuous functions converges absolutely converges to f(x0 converges to xo converges to zero converges uniformly countable set Define f differentiable at x0 diverges Exercise f and g f dx f is continuous f is differentiable f is uniformly fact finite number following theorem function f hence implies infinite series irrational number lemma Let f lim f(x limit at x0 limit at zero lower bound mathematical induction Mean-Value Theorem monotone neighborhood nonempty number of terms partial sum partition polynomial positive integer positive real number power series PROJECT Proof Suppose Prove that f radius of convergence rational numbers reader Riemann-integrable sequence converges series converges set of real show that f subsequence subset Suppose f Taylor's Theorem Theorem 6.8 THEOREM Let THEOREM Suppose uniformly continuous well-ordering principle