Introduction to AnalysisAmerican Mathematical Soc., 2009 - 240 páginas 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. |
Contenido
Preliminaries | 2 |
Sequences | 33 |
Chapter2 Limits of Functions | 63 |
Continuity | 83 |
chapter4 Differentiation III | 111 |
The Riemann Integral | 137 |
Infinite Series | 173 |
Sequences | 215 |
237 | |
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accumulation point Assume belong bounded called Cauchy Chapter Choose clear closed compact conclude condition Consider contains continuous at xo converges absolutely converges to xo converges to zero converges uniformly countable course Define Define f definition derivative determine differentiable diverges equal equivalent Example Exercise exists f dx f is continuous f is differentiable fact finite function f give given graph hence idea implies increasing induction infinite series interval irrational least lemma limit limit at x0 mathematical means neighborhood Note observe obtain partition polynomial positive integer power series preceding PROJECT proof Prove Prove that f radius of convergence rational numbers reader real number result rule satisfy sequence statement subsequence subset Suppose f THEOREM Let THEOREM Suppose true uniformly continuous upper bound write