Analysis: With an Introduction to ProofFor courses in Real Analysis, Advanced Calculus, and Transition to Advanced Mathematics or Proofs course. Carefully focused on reading and writing proofs, this introduction to the analysis of functions of a single real variable helps students in the transition from computationally oriented courses to abstract mathematics by its emphasis on proofs. Student oriented and instructor friendly, it features clear expositions and examples, helpful practice problems, many drawings that illustrate key ideas, and hints/answers for selected exercises. *NEW - True/False questions - (More than 250 total) located at the beginning of the exercises for each section and relating directly to the reading. *NEW - 8 new illustrations of key concepts make this the most visually compelling analysis text. *Straightforward discussion of logic - As it applies to the proving of theorems in analysis (Ch. 1). Can be covered briefly or in depth, depending on the needs of students. *Practice problems - Scattered throughout the narrative (more than 140 total). These problems relate directly to what has just been presented. Includes complete answers at the end of each section. *Fill-in-the-blank proofs. Helps stude |
From inside the book
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Page 11
... positive number x such that x2 = 5 . ( b ) For every positive number M there is a positive number N such that N < 1 / M . ( c ) If n≥ N , then [ fn ( x ) f ( x ) | ≤ 3 for all x in A. Having seen several examples of how existential ...
... positive number x such that x2 = 5 . ( b ) For every positive number M there is a positive number N such that N < 1 / M . ( c ) If n≥ N , then [ fn ( x ) f ( x ) | ≤ 3 for all x in A. Having seen several examples of how existential ...
Page 23
... number , then n is an even number . ( Use the fact that a number is odd iff ... positive integers . In Example 3.7 when talking about odd and even numbers ... positive numbers but is false when considering all real numbers . Similarly ...
... number , then n is an even number . ( Use the fact that a number is odd iff ... positive integers . In Example 3.7 when talking about odd and even numbers ... positive numbers but is false when considering all real numbers . Similarly ...
Page 24
... positive numbers and true for reals . When we introduce set notation ( in Chapter 2 ) , it will become easier to be ... number ɛ . Thus we begin by letting & be an arbitrary positive number . We need to use this & to find a positive ...
... positive numbers and true for reals . When we introduce set notation ( in Chapter 2 ) , it will become easier to be ... number ɛ . Thus we begin by letting & be an arbitrary positive number . We need to use this & to find a positive ...
Contents
Sets and Functions | 31 |
The Real Numbers | 87 |
Ordered Fields | 95 |
Copyright | |
22 other sections not shown
Common terms and phrases
a₁ accumulation point ANSWERS TO PRACTICE axiom of choice bijection Cauchy sequence conclude continuous function converges uniformly Corollary countable definition denote denumerable derivative diverges equivalence relation Exercise exists a neighborhood exists a number exists a point Figure finite function defined function f ƒ and g ƒ is continuous ƒ is differentiable ƒ is integrable implies induction inequality infinite injective interval of convergence Justify each answer Let f Let f(x Let ƒ lim sup Mark each statement mean value theorem metric space N₁ natural numbers obtain open set ordered field partial sums partition polynomial positive number power series PRACTICE PROBLEMS Proof properties Prove that ƒ radius of convergence rational numbers real number s₁ Section series converges statement True subsequential limits Suppose that ƒ surjective True or False uniform convergence uniformly continuous upper bound