Introduction to Real AnalysisIn recent years, mathematics has become valuable in many areas, including economics and management science as well as the physical sciences, engineering and computer science. Therefore, this book provides the fundamental concepts and techniques of real analysis for readers in all of these areas. It helps one develop the ability to think deductively, analyze mathematical situations and extend ideas to a new context. Like the first two editions, this edition maintains the same spirit and user-friendly approach with some streamlined arguments, a few new examples, rearranged topics, and a new chapter on the Generalized Riemann Integral. |
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Page 3
... elements is called the empty set and is denoted by the symbol Ø . Two sets A and B are said to be disjoint if they have no elements in common ; this can be expressed by writing AB = Ø . To illustrate the method of proving set equalities ...
... elements is called the empty set and is denoted by the symbol Ø . Two sets A and B are said to be disjoint if they have no elements in common ; this can be expressed by writing AB = Ø . To illustrate the method of proving set equalities ...
Page 17
... elements . Further , a set T1 is finite if and only if there is a bijection from T1 onto another set T2 that is finite . It is now necessary to establish some basic properties of finite sets to be sure that the definitions do not lead ...
... elements . Further , a set T1 is finite if and only if there is a bijection from T1 onto another set T2 that is finite . It is now necessary to establish some basic properties of finite sets to be sure that the definitions do not lead ...
Page 23
... element 0 in R such that 0 + a = a and a +0 = a for all a in R ( existence of a zero element ) ; ( A4 ) for each a in R there exists an element -a in R such that a + ( -a ) = 0 and ( -a ) + a = 0 ( existence of negative elements ) ...
... element 0 in R such that 0 + a = a and a +0 = a for all a in R ( existence of a zero element ) ; ( A4 ) for each a in R there exists an element -a in R such that a + ( -a ) = 0 and ( -a ) + a = 0 ( existence of negative elements ) ...
Contents
PRELIMINARIES | 1 |
THE REAL NUMBERS | 22 |
SEQUENCES AND SERIES | 52 |
Copyright | |
12 other sections not shown
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8-fine partition a₁ absolutely convergent apply arbitrary b₁ belongs to R*[a bijection calculation Cauchy sequence cluster point conclude continuous functions Convergence Theorem converges uniformly countable defined derivative differentiable divergent endpoint Exercises for Section exists finite number follows from Theorem function f Fundamental Theorem ƒ and g ƒ is continuous gauge Hence implies increasing sequence infinite inverse Let f Let f(x let ƒ lim f lim f(x lim ƒ lim(x limit Mathematical Induction Mean Value Theorem metric space monotone natural number neighborhood nonempty obtain open interval open set P₁ P₂ partial sums properties prove rational numbers reader real numbers result Riemann integrable S₁ satisfies sequence of real show that ƒ Squeeze Theorem step function subintervals subset Suppose that ƒ supremum tagged partition Taylor's Theorem Theorem Let Triangle Inequality uniform convergence uniformly continuous upper bound x₁