PureMath.pdf

# On the other hand the class of positive integers is

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On the other hand the class of positive integers is not finite but infinite. This may be expressed more precisely as follows. If n is any positive integer, such as 1000, 1 , 000 , 000 or any number we like to think of, then there are more than n positive integers. Thus, if the number we think of is 1 , 000 , 000, there are obviously at least 1 , 000 , 001 positive integers. Similarly the class of rational numbers, or of real numbers, is infinite. It is convenient to express this by saying that there are an infinite number of positive integers, or rational numbers, or real numbers. But the reader must be careful always to remember that by saying this we mean simply that the class in question has not a finite number of members such as 1000 or 1 , 000 , 000. 53. Properties possessed by a function of n for large values of n . We may now return to the ‘functions of n ’ which we were discussing in §§ 50 51 . They have many points of difference from the functions of x which we discussed in Chap. II . But there is one fundamental character- istic which the two classes of functions have in common: the values of the variable for which they are defined form an infinite class . It is this fact which forms the basis of all the considerations which follow and which, as we shall see in the next chapter, apply, mutatis mutandis , to functions of x as well. Suppose that φ ( n ) is any function of n , and that P is any property which φ ( n ) may or may not have, such as that of being a positive integer

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[IV : 53] LIMITS OF FUNCTIONS OF A 132 or of being greater than 1. Consider, for each of the values n = 1, 2, 3, . . . , whether φ ( n ) has the property P or not. Then there are three possibilities:— ( a ) φ ( n ) may have the property P for all values of n , or for all values of n except a finite number N of such values: ( b ) φ ( n ) may have the property for no values of n , or only for a finite number N of such values: ( c ) neither ( a ) nor ( b ) may be true. If ( b ) is true, the values of n for which φ ( n ) has the property form a finite class. If ( a ) is true, the values of n for which φ ( n ) has not the property form a finite class. In the third case neither class is finite. Let us consider some particular cases. (1) Let φ ( n ) = n , and let P be the property of being a positive integer. Then φ ( n ) has the property P for all values of n . If on the other hand P denotes the property of being a positive integer greater than or equal to 1000, then φ ( n ) has the property for all values of n except a finite number of values of n , viz. 1, 2, 3, . . . , 999. In either of these cases ( a ) is true. (2) If φ ( n ) = n , and P is the property of being less than 1000, then ( b ) is true. (3) If φ ( n ) = n , and P is the property of being odd, then ( c ) is true. For φ ( n ) is odd if n is odd and even if n is even, and both the odd and the even values of n form an infinite class. Example. Consider, in each of the following cases, whether ( a ), ( b ), or ( c ) is true: (i) φ ( n ) = n , P being the property of being a perfect square, (ii) φ ( n ) = p n , where p n denotes the n th prime number, P being the property of being odd, (iii) φ ( n ) = p n , P being the property of being even, (iv) φ ( n ) = p n , P being the property φ ( n ) > n , (v) φ ( n
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