# An Introduction to Real Analysis by Derek G. Ball and C. Plumpton (Auth.) PDF

By Derek G. Ball and C. Plumpton (Auth.)

ISBN-10: 0080169368

ISBN-13: 9780080169361

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Extra resources for An Introduction to Real Analysis

Sample text

The symbol is only to be used in conjunction with the idea of a limit, to have precisely the meaning described by the definition. 40 AN INTRODUCTION TO REAL ANALYSIS It is, of course, important to be able to use the definition above to show that a particular series tends to infinity. For the moment we give one easy example to illustrate this. Consider the sequence 1,4,9, 16,25,36,49,64, . . Here an = n2. If we are given K = 1000 we may choose N = 32, and then an ^ a32 = 1024 > 1000 whenever n Ξ> 32.

Irrationals Those real numbers which are not rationals are called irrationals. We have already seen that \/2 is irrational. So also is \/n, where n is any natural number other than a perfect square ; \/n, where n is any natural number other than a perfect cube. So also are a variety of totally different numbers like π, e, and 2V2, although it is sometimes very difficult to prove that a number is irrational. Purely as examples we prove two results. (1) \/12 is irrational. Suppose %/12 is rational, so that \/\2 = p/q, where p and q are natural numbers with no common factor.

For if fl|j = ( - 1 ) 7 « then lim Λ„ = 0. But £„ = {-\)nn and so {£„} oscillates. 6. {an} is a sequence and lim an — I. (i) If for all n, an^ M then / M\ then /—M > 0. Since lim α„ = /, there is a natural number TV such that for all n ^ TV, | αΛ—1\ < /—Af. Thus for all n^ N,an > M. This contradicts the conditions of the theorem. The proof of (ii) is similar and is left as an exercise. D COROLLARY. {an} and {bn} are sequences, and lim an = /, lim bn = m.