By Steven R. Lay

Research with an creation to evidence, 5th variation is helping fill within the foundation scholars have to achieve genuine analysis-often thought of the main tricky direction within the undergraduate curriculum. by means of introducing common sense and emphasizing the constitution and nature of the arguments used, this article is helping scholars circulate conscientiously from computationally orientated classes to summary arithmetic with its emphasis on proofs. transparent expositions and examples, worthwhile perform difficulties, a number of drawings, and chosen hints/answers make this article readable, student-oriented, and instructor- pleasant. 1. good judgment and facts 2. units and services three. the genuine Numbers four. Sequences five. Limits and Continuity 6. Differentiation 7. Integration eight. endless sequence Steven R. Lay word list of keyword phrases Index

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If 1 ∫ 0 f ( x) dx ≠ 0, then there exists a point x in the interval [0, 1] such that f (x) ≠ 0. Symbolically, we have p ⇒ q, where p: 1 ∫ 0 f ( x) dx ≠ 0 q : ∃ x in [0, 1] f (x) ≠ 0. The contrapositive implication, ~ q ⇒ ~ p, can be written as If for every x in [0, 1], f ( x) = 0, then 1 ∫ 0 f ( x) = 0. This is much easier to prove. Instead of having to conclude the existence of an x in [0, 1] with a particular property, we are given that every x in [0, 1] has a different property. Indeed, the proof now follows directly from the definition of the integral, since each of the terms in any upper or lower Riemann sum will be zero.

Then {Ex : x ∈ S} is a partition of S. The relation “belongs to the same piece as” is the same as R. Conversely, if p is a partition of S, let P be defined by x P y iff x and y are in the same piece of the partition. Then P is an equivalence relation and the corresponding partition into equivalence classes is the same as p. Sets and Functions Proof: Let R be an equivalence relation on S. We have already shown that {Ex : x ∈ S} is a partition. ” Then xP y iff x, y ∈ Ez for some z ∈ S iff x R z and y R z for some z ∈ S iff x R y.

6 PRACTICE Let A = {1, 2, 3, 4, 5} B = {x : x = 2k for some k ∈ N} C = {x ∈ N : x < 6}. Which of the following statements are true? 2(d) we found that the collection D of all prime numbers between 8 and 10 is a legitimate set. This is so because the statement “ x ∈ D ” is always false, since there are no prime numbers between 8 and 10. Thus D is an example of the empty set, a set with no members. It is not difficult to show (Exercise 18) that there is only one empty set, and we denote it by ∅. For our first theorem we shall prove that the empty set is a subset of every set.