By Graeme L. Cohen

ISBN-10: 0511061668

ISBN-13: 9780511061660

ISBN-10: 0511070128

ISBN-13: 9780511070129

Designed for one-semester classes for senior undergraduates, this booklet ways themes at the beginning via convergence of sequences in metric area. in spite of the fact that, the choice topological technique is additionally defined. purposes are integrated from differential and necessary equations, platforms of linear algebraic equations, approximation idea, numerical research and quantum mechanics.

Cover; Half-title; Series-title; name; Copyright; Contents; Preface; 1 Prelude to trendy research; 2 Metric areas; three The fastened aspect Theorem and its functions; four Compactness; five Topological areas; 6 Normed Vector areas; 7 Mappings on Normed areas; eight internal Product areas; nine Hilbert house; Bibliography; chosen ideas; Index.

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The example above of a sequence of rational numbers converging to an irrational number shows that this too is not a property of the system of rational numbers alone. 12 (C au ch y C on v ergen ce C rite rion ) A sequence {a n} is convergent if and only if for any number e > 0 there exists a positive integer N such that |&n — &m| < e whenever m , n > N. It is easy to see that the condition is necessary. To do so, suppose { a n} is convergent and lim an = £. 5 that a positive integer N exists such that \an — £| < |e whenever n > N.

Retrieving the pieces is technically described as adding more and more structure to the elements of the sets: allowing the notion of a distance between pairs of elements, allowing the elements to be able to be added together, and so on. Each phase determines what is known as a space. It is not required that the elements of any of these spaces (ex­ cept perhaps the ultimate one) actually be real numbers, but just that they have properties suggested by certain properties of real numbers. This explains why up to now, and particularly in the preceding two sections, we have concentrated on properties of real numbers.

This proves (b), a result which is often expressed by saying: the union of count ably many countable sets is itself a countable set. □ It should be clear that the proof of (b) covers the cases where there are only finitely many sets Xk, and where some of these are finite sets. In particular, it implies that the union o f two countable sets is countable. We now prove two fundamental results. 3 (a) The set Q of rational numbers is countable. (b) The set R of real numbers is uncountable. To prove (a), for each k E N let Xk be the set of all rational numbers that can be expressed as pf k where p E Z.

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A course in modern analysis and its applications by Graeme L. Cohen

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