A Concrete Introduction to Real Analysis

Robert Carlson

• Éditeur: CRC Press
• Année: 2017
• ISBN:
  9781498778152
• Édition: 2
• Sujet: Mathématiques
• Sous-sujet: Mathématiques générales

A Concrete Introduction to Analysis, Second Edition offers a major reorganization of the previous edition with the goal of making it a much more comprehensive and accessible for students.

The standard, austere approach to teaching modern mathematics with its emphasis on formal proofs can be challenging and discouraging for many students. To remedy this situation, the new edition is more rewarding and inviting. Students benefit from the text by gaining a solid foundational knowledge of analysis, which they can use in their fields of study and chosen professions.

The new edition capitalizes on the trend to combine topics from a traditional transition to proofs course with a first course on analysis. Like the first edition, the text is appropriate for a one- or two-semester introductory analysis or real analysis course. The choice of topics and level of coverage is suitable for mathematics majors, future teachers, and students studying engineering or other fields requiring a solid, working knowledge of undergraduate mathematics.

Key highlights:

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• Offers integration of transition topics to assist with the necessary background for analysis

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• Can be used for either a one- or a two-semester course

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• Explores how ideas of analysis appear in a broader context

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• Provides as major reorganization of the first edition

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• Includes solutions at the end of the book

1

Real numbers and mathematical proofs

Simple facts of arithmetic or geometry are often tested by common experience. If you throw six nuts into a basket, and then add ten more, you get the same total as if ten went in first, followed by six. The commutativity of addition is thus testable in a meaningful way.

The same cannot be said for many of the results of mathematics. What direct experience suggests that there are infinitely many prime numbers, or that the square root of two is not the quotient of two integers? Is the scarcity of solutions to the equation

$x^n+y^n=z^n$

convincing evidence that this equation has no positive integer solutions if n is an integer bigger than 2?

Experience can also mislead. We can easily see 6 nuts, but finding –5 nuts is more challenging; even professional mathematicians had trouble with negative numbers through the eighteenth century [5, p. 592-3]. Familiarity with integers, rational numbers, and a geometric understanding of real numbers as lengths makes it hard to believe that there is a number whose square is –1.

How then do we develop sound mathematical reasoning? Statements, ideas, or algorithms need not be valid simply because they are phrased in a precise way, have been tested in a few cases, or appeal to common intuition. In times past experts believed that every length could be represented as the ratio of two integers, and that squares of numbers are necessarily greater than or equal to 0. Brilliant minds believed, along with most calculus students, that except for isolated exceptional points, all functions have derivatives of all orders at every point. Even professional logicians seemed unwary of the traps in statements such as “there is a barber i...