Introduction to real analysis
Recognizing the increased role of real analysis in economics, management engineering and computer science as well as in the physical sciences, this Second Edition meets the need for an accessible, comprehensive textbook regarding the fundamental concepts and techniques in this area of mathematics. Provides solid coverage of real analysis fundamentals with an emphasis on topics from numerical analysis and approximation theory because of their increased importance to contemporary students. Topics include real numbers, sequences, limits, continuous functions, differentiation, infinite series and more. Topological concepts are now conveniently combined into one chapter. An appendix on logic and proofs helps students in analyzing proofs of theorems.
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CHAPTER TWO The Real Numbers
CHAPTER THREE Sequences
CHAPTER FOUR Limits
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absolutely convergent apply approximation arbitrary Archimedean Property bounded function calculation Cauchy sequence cluster point compact conclude continuous functions convergent sequence converges uniformly Corollary countable decimal defined Definition Let denote derivative differentiable divergent E(xn elements end point establish Exercises for Section Figure finite number follows from Theorem function g g is continuous given Hence implies improper integral inequality infinite injective inverse Lemma let f Let f(x lim(xn limit mathematical induction Mean Value Theorem metric space monotone natural number neighborhood nonempty notation Note obtain open interval open set partition polynomial Proof properties Prove rational number reader real numbers result Riemann sum satisfy sequence of real sequence xn Show that lim statement strictly increasing subinterval subsequence subset Suppose supremum Taylor's Theorem Theorem Let uniform convergence uniformly continuous upper bound Vs(c whence it follows