# New PDF release: An Introduction to Real Analysis

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

ISBN-10: 0080169368

ISBN-13: 9780080169361

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**Sample text**

Then for ail n ^ P, 2£ ' '2|m| + l Therefore lim en = Im. (iv) Since lim an = I and / ^ O , there is a natural number iV with K - / | < 1/1/2 for all « ^ AT. Now \l-an\ + \an\^ \l-an+an\ = \l\ so that \an\ > | / | - | / | / 2 = | / | / 2 for all n^N. Given e > 0, there is a natural number M with 10rt—/| < el2/2 for all η^ Μ. Put P = max(#, M). |/||«„| ε * Therefore lim/„ = 1//. D COROLLARY. {an} and {bn} are sequences with lim an = /, lim bn = m. £ 0. Proof. We write gn = an»l/bn and use parts (iii) and (iv) of the Theorem.

SEQUENCES 39 3. {an} and {bn} are decreasing sequences and the sequences {cn} and {dn} are defined by cn = an+bni dn = anbn. (i) Is {cn} necessarily decreasing? (ii) Is {dn} necessarily decreasing? Justify your answers. 2. Limits of sequences Sequences generally arise as sets of closer and closer approximations to some real number. The limit of a sequence is a formulation of this idea. Consider the sequences 1,4,9, 16,25,36, . . , 2, 6, 4, 8, 10, 8, 12, . . The first is an increasing sequence; the second is not.

Here we discuss one of the most obvious and most important ways. An increasing sequence {an} is a sequence for which DEFINITION. an^^anforalln. DEFINITION. A strictly increasing sequence {an} is a sequence for which an+1 > anfor all n. A decreasing sequence {an} is a sequence for which DEFINITION. an+1^anforalln. DEFINITION. which an+1< A strictly decreasing sequence {an} is a sequence for anfor all n. 38 AN INTRODUCTION TO REAL ANALYSIS DEFINITION. A monotonie sequence is a sequence which is either increasing or decreasing.

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

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