Mathematical Analysis

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

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Extra info for An Introduction to Real Analysis

Example text

Given two real numbers x — (Li, Ri) and y = (L2, R2) we say that y is greater than xifLi is a proper subset ofL2. ) If we have two real numbers (Li, R{) and (L 2 , ^2) it may be readily shown by use of the definition of a Dedekind section that L\ c L 2 or L± = L2 or L\ Z) L 2 . Thus x < y or x — y or x > y. If we are to use the real numbers as a generalization of the rationals, we shall require a meaning for the addition and multiplication of real numbers. Although the ideas behind the definitions are obvious, the details of relevant definitions and proofs are tedious,so we shall omit them here.

The proof that the other properties of a Dedekind section are satisfied by (L*, R*) is left as an exercise. DEFINITION (the product of x and y if at least one of x and y is negative). Ifx is positive and y is negative then x and —y are positive so that X' —y is defined. We define xy = — (x· —y). So, too, if x is negative and y is positive, we define xy = —(—x*y) and, if x and y are both negative, we define xy = — x* —y. 6 1. If x, y9 and z are real numbers show (i) x+y = y+x; (ii) (x+y)+z = x+(y+z).

A sequence {an} tends to infinity as n tends to infinity (written a -*■ °o as n -*■ °°) if given any real number K, there exists a natural number N (which generally depends upon the choice of K) such that an > Kfor all η^ Ν. If an -*~ °° as « -^ co we also write lim an = <*> or even simply lim an = oo. The reader should note carefully that our use of the symbol o° does not imply the existence of a number called infinity. The symbol is only to be used in conjunction with the idea of a limit, to have precisely the meaning described by the definition.

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