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 44
... endpoints of the interval ; however , the endpoints are not included in an open interval . If both endpoints are adjoined to this open interval , then we obtain the closed interval determined by a and b ; namely , the set [ a , b ] ...
... endpoints of the interval ; however , the endpoints are not included in an open interval . If both endpoints are adjoined to this open interval , then we obtain the closed interval determined by a and b ; namely , the set [ a , b ] ...
Page 45
... endpoints gives us the infinite closed intervals : [ a , ∞ ) : = { x ЄR : a ≤ x } and ( -∞ , b ] : = { x ЄR : x ... endpoint of ( -∞ , ∞ ) . Warning It must be emphasized that ∞ and -∞ are not elements of R , but only conve- nient ...
... endpoints gives us the infinite closed intervals : [ a , ∞ ) : = { x ЄR : a ≤ x } and ( -∞ , b ] : = { x ЄR : x ... endpoint of ( -∞ , ∞ ) . Warning It must be emphasized that ∞ and -∞ are not elements of R , but only conve- nient ...
Page 150
... endpoint of the interval on which ƒ is defined . 5.6.2 Corollary Let ICR be an interval and let f : I → R be increasing on I. Suppose that cЄ I is not an endpoint of I. Then the following statements are equivalent . C ( a ) f is ...
... endpoint of the interval on which ƒ is defined . 5.6.2 Corollary Let ICR be an interval and let f : I → R be increasing on I. Suppose that cЄ I is not an endpoint of I. Then the following statements are equivalent . C ( a ) f is ...
Contents
PRELIMINARIES | 1 |
THE REAL NUMBERS | 22 |
SEQUENCES AND SERIES | 52 |
Copyright | |
12 other sections not shown
Common terms and phrases
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