Introduction to Real Analysis

Introduction to Real Analysis

Language: English

Pages: 416

ISBN: 0471433314

Format: PDF / Kindle (mobi) / ePub

This text provides the fundamental concepts and techniques of real analysis for students in all of these areas. It helps one develop the ability to think deductively, analyse mathematical situations and extend ideas to a new context. Like the first three editions, this edition maintains the same spirit and user-friendly approach with addition examples and expansion on Logical Operations and Set Theory. There is also content revision in the following areas: introducing point-set topology before discussing continuity, including a more thorough discussion of limsup and limimf, covering series directly following sequences, adding coverage of Lebesgue Integral and the construction of the reals, and drawing student attention to possible applications wherever possible.

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casual observer to conclude that the sequence is bounded. However, the sequence is in fact divergent, which is established by noting that     1 1 1 1 1 þ h2 n ¼ 1 þ þ þ Á Á Á þ nÀ1 þ ÁÁÁ þ n 2 3 4 2 2 þ1     1 1 1 1 1 þ > 1þ þ þ ÁÁÁ þ n þ ÁÁÁ þ n 2 4 4 2 2 1 1 1 ¼ 1 þ þ þ ÁÁÁ þ 2 2 2 n ¼ 1þ : 2 Since ðhn Þ is unbounded, Theorem 3.2.2 implies that it is divergent. (This proves that the infinite series known as the harmonic series diverges. See Example 3.7.6(b) in Section 3.7.) The terms hn

converges. (It is far from obvious that the limit of this series is equal to ln 2.) Comparison Tests Our first test shows that if the terms of a nonnegative series are dominated by the corresponding terms of a convergent series, then the first series is convergent. 3.7.7 Comparison Test Let X :¼ (xn) and Y :¼ (yn) be real sequences and suppose that for some K 2 N we have (8) 0 xn yn for n ! K: P P (a) Then the convergence of yn implies the convergence of xn . P P (b) The divergence of xn

conclude that the sequence (xn) in An{c} converges to c, but the sequence ð f ðxn ÞÞ does not converge to L. Therefore we have shown that if (i) is not true, then (ii) is not true. We Q.E.D. conclude that (ii) implies (i). We shall see in the next section that many of the basic limit properties of functions can be established by using corresponding properties for convergent sequences. For example, we know from our with sequences that if (xn) is any sequence that converges to a À work Á number c,

continuous function that does not take on any of its values twice and with f (0) < f (1). Show that f is strictly increasing on [0, 1]. 13. Let h : ½0; 1Š ! R be a function that takes on each of its values exactly twice. Show that h cannot be continuous at every point. [Hint: If c1 < c2 are the points where h attains its supremum, show that c1 ¼ 0, c2 ¼ 1. Now examine the points where h attains its infimum.] À Ám À Áp 14. Let x 2 R, x > 0. Show that if m, p 2 Z, n, q 2 N, and mq ¼ np, then x1=n ¼

The situation for strictly decreasing functions is similar. Remark It is reasonable to define a function to be increasing at a point if there is a neighborhood of the point on which the function is increasing. One might suppose that, if the derivative is strictly positive at a point, then the function is increasing at this point. However, this supposition is false; indeed, the differentiable function defined by & x þ 2x2 sin ð1=xÞ if x 6¼ 0; gðxÞ :¼ 0 if x ¼ 0; is such that g0 ð0Þ ¼ 1, yet it can

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