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 335
... statements : the first is true , the second is false , and the third is either true or false , but we are not sure which at this time . The fourth sentence is not a statement ; it can be neither true nor false since it leads to ...
... statements : the first is true , the second is false , and the third is either true or false , but we are not sure which at this time . The fourth sentence is not a statement ; it can be neither true nor false since it leads to ...
Page 336
... statements P : x = 2 , Q : У є А. = The statement ( P and Q ) is true when both ( x 2 ) and ( y Є A ) are true , and it is false when at least one of ( x = 2 ) and ( y e A ) is false ; that is , the statement not ( P and Q ) is true ...
... statements P : x = 2 , Q : У є А. = The statement ( P and Q ) is true when both ( x 2 ) and ( y Є A ) are true , and it is false when at least one of ( x = 2 ) and ( y e A ) is false ; that is , the statement not ( P and Q ) is true ...
Page 338
... statements For any integer x , x2 = 1 and There exists an integer x such that x2 = 1 . Clearly the first statement is false , as is seen by taking x = 3 ; however , the second statement is true since we can take either x = 1 or x = -1 ...
... statements For any integer x , x2 = 1 and There exists an integer x such that x2 = 1 . Clearly the first statement is false , as is seen by taking x = 3 ; however , the second statement is true since we can take either x = 1 or x = -1 ...
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 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 inverse Let A CR 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 partial sums properties prove rational numbers reader real numbers result Riemann integrable S₁ S₂ satisfies sequence of real show that ƒ Squeeze Theorem step functions subintervals subset Suppose that ƒ supremum tagged partition Taylor's Theorem Theorem Let Triangle Inequality uniform convergence uniformly continuous upper bound x₁