PureMath.pdf

# For all values of n ii if is any positive number then

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for all values of n ; (ii) if is any positive number then φ ( n ) > M - for at least one value of n . This number M we call the upper bound of φ ( n ). Similarly, if φ ( n ) is bounded below, that is to say if there is a number k such that φ ( n ) 5 k for all values of n , then there is a number m such that (i) φ ( n ) = m for all values of n ; (ii) if is any positive number then φ ( n ) < m + for at least one value of n . This number m we call the lower bound of φ ( n ). If K exists, M 5 K ; if k exists, m = k ; and if both k and K exist then k 5 m 5 M 5 K. 82. The limits of indetermination of a bounded function. Sup- pose that φ ( n ) is a bounded function, and M and m its upper and lower bounds. Let us take any real number ξ , and consider now the relations of inequality which

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[IV : 82] POSITIVE INTEGRAL VARIABLE 181 may hold between ξ and the values assumed by φ ( n ) for large values of n . There are three mutually exclusive possibilities: (1) ξ = φ ( n ) for all sufficiently large values of n ; (2) ξ 5 φ ( n ) for all sufficiently large values of n ; (3) ξ < φ ( n ) for an infinity of values of n , and also ξ > φ ( n ) for an infinity of values of n . In case (1) we shall say that ξ is a superior number, in case (2) that it is an inferior number, and in case (3) that it is an intermediate number. It is plain that no superior number can be less than m , and no inferior number greater than M . Let us consider the aggregate of all superior numbers. It is bounded below, since none of its members are less than m , and has therefore a lower bound, which we shall denote by Λ. Similarly the aggregate of inferior numbers has an upper bound, which we denote by λ . We call Λ and λ respectively the upper and lower limits of indetermination of φ ( n ) as n tends to infinity ; and write Λ = lim φ ( n ) , λ = lim φ ( n ) . These numbers have the following properties: (1) m 5 λ 5 Λ 5 M ; (2) Λ and λ are the upper and lower bounds of the aggregate of intermediate numbers, if any such exist; (3) if is any positive number, then φ ( n ) < Λ + for all sufficiently large values of n , and φ ( n ) > Λ - for an infinity of values of n ; (4) similarly φ ( n ) > λ - for all sufficiently large values of n , and φ ( n ) < λ + for an infinity of values of n ; (5) the necessary and sufficient condition that φ ( n ) should tend to a limit is that Λ = λ , and in this case the limit is l , the common value of λ and Λ. Of these properties, (1) is an immediate consequence of the definitions; and we can prove (2) as follows. If Λ = λ = l , there can be at most one intermediate number, viz. l , and there is nothing to prove. Suppose then that Λ > λ . Any intermediate number ξ is less than any superior and greater than any inferior number, so that λ 5 ξ 5 Λ. But if λ < ξ < Λ then ξ must be intermediate, since it is plainly neither superior nor inferior. Hence there are intermediate numbers as near as we please to either λ or Λ.
[IV : 82] LIMITS OF FUNCTIONS OF A 182 To prove (3) we observe that Λ + is superior and Λ - intermediate or inferior. The result is then an immediate consequence of the definitions; and the proof of (4) is substantially the same.

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