By Donald Yau

ISBN-10: 9814417858

ISBN-13: 9789814417853

This booklet is an introductory textual content on genuine research for undergraduate scholars. The prerequisite for this publication is an exceptional historical past in freshman calculus in a single variable. The meant viewers of this booklet contains undergraduate arithmetic majors and scholars from different disciplines who use actual research. due to the fact this publication is aimed toward scholars who would not have a lot past adventure with proofs, the speed is slower in prior chapters than in later chapters. There are thousands of workouts, and tricks for a few of them are incorporated.

Readership: Undergraduates and graduate scholars in research.

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**Extra resources for A First Course in Analysis**

**Sample text**

So let > 0 be given. Since α − < α, α − is not an upper bound of the set A. In other words, there exists an aN such that α − < aN . Then for n ≥ N , we have α − < aN ≤ an ≤ α < α + , from which we obtain ∣an − α∣ < . This proves that lim an = α. 5in 40 A First Course in Analysis The reader should be careful that the Monotone Convergence Theorem does not assert that a convergent sequence is monotone. 10. The sequence {(−1)n n1 } = {−1, 21 , − 13 , 41 , . } converges to 0. However, it is neither increasing nor decreasing.

Then it is a convergent sequence. Therefore, a sequence is convergent if and only if it is a Cauchy sequence. Proof. This is again an 2 -argument. 8 implies that it has a convergent subsequence {ank }. Let L be the limit of this subsequence. We will show that lim an = L. Given > 0, since lim ank = L, there exists a positive integer nK such that nk ≥ nK implies ∣ank − L∣ < . 2 Since {an } is a Cauchy sequence, there exists a positive integer N ≥ nK such that n, m ≥ N implies ∣an − am ∣ < . 2 For integers m ≥ N , we have nm ≥ N ≥ nK .

Then T is a finite set. Proof. 8. Indeed, if T is an infinite set, then S is an infinite set as well, contradicting the assumption. 7. The sets Z, Q, and R are all infinite. Proof. 9 that N is an infinite set. 8 that they are also infinite sets. 2 Countable and Uncountable Sets What is perhaps surprising is that there are very different sizes even among infinite sets. To make this precise, we need the following definitions. 7. An infinite set S is said to be countable if there is a bijection from Z+ to S.

### A First Course in Analysis by Donald Yau

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