By Steven R. Lay

Research with an advent to facts, 5th version is helping fill within the basis scholars have to reach actual analysis-often thought of the main tough path within the undergraduate curriculum. by means of introducing good judgment and emphasizing the constitution and nature of the arguments used, this article is helping scholars movement rigorously from computationally orientated classes to summary arithmetic with its emphasis on proofs. transparent expositions and examples, necessary perform difficulties, a number of drawings, and chosen hints/answers make this article readable, student-oriented, and instructor- pleasant. 1. common sense and facts 2. units and services three. the true Numbers four. Sequences five. Limits and Continuity 6. Differentiation 7. Integration eight. endless sequence Steven R. Lay word list of key words Index

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21. Prove or give a counterexample: A \(A \B) = B \(B \A). 22. Prove or give a counterexample: A \(B \A) = B \(A \B). 23. Let (a) (b) (c) A and B be subsets of a universal set U. Prove the following. A\B = (U \B) \(U \A) U \(A\B) = (U \A) ∪ B (A\B) ∪ (B \A) = (A ∪ B) \(A ∩ B) 24. 13. 25. Find U B∈b B and I B∈b B for each collection b. ⎧⎡ 1⎤ ⎫ (a) b = ⎨ ⎢1, 1 + ⎥ : n ∈ N ⎬ n⎦ ⎩⎣ ⎭ ⎧⎛ 1⎞ ⎫ (b) b = ⎨⎜ 1, 1 + ⎟ : n ∈ N ⎬ n⎠ ⎩⎝ ⎭ (c) b = {[2, x] : x ∈ R and x > 2} (d) b = {[0,3], (1,5), [2, 4)} *26.

If x ∈ A, then __________ ________________________________________________________. Thus A ⊆ B. ♦ 12. Suppose you are to prove that set A is a subset of set B. Write a reasonable beginning sentence for the proof, and indicate what you would have to show in order to finish the proof. 13. Suppose you are to prove that sets A and B are disjoint. Write a reasonable beginning sentence for the proof, and indicate what you would have to show in order to finish the proof. 14. Which statement(s) below would enable one to conclude that x ∈ A ∪ B?

But before we close this chapter on logic and proof, a few informal comments are in order. ) be very careful to use sound logical reasoning. This is what we have tried to help you develop in this chapter. But when writing down a proof it is usually unnecessary – and often undesirable – to include all the logical steps and details along the way. The human mind can only absorb so much information at one time. It is necessary to skip lightly over the steps that are well understood from previous experience so that greater attention can be focused on the part that is really new.

### Analysis with an introduction to proof by Steven R. Lay

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