Their behaviour as n tends to this apparent

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their behaviour as n tends to , this apparent incongruity in the definition is not a serious defect. There is one exceedingly important theorem concerning functions of this class. Theorem. If φ ( n ) steadily increases with n , then either (i) φ ( n ) tends to a limit as n tends to , or (ii) φ ( n ) + . That is to say, while there are in general five alternatives as to the behaviour of a function, there are two only for this special kind of function.
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[IV : 69] LIMITS OF FUNCTIONS OF A 158 This theorem is a simple corollary of Dedekind’s Theorem ( § 17 ). We divide the real numbers ξ into two classes L and R , putting ξ in L or R according as φ ( n ) = ξ for some value of n (and so of course for all greater values), or φ ( n ) < ξ for all values of n . The class L certainly exists; the class R may or may not. If it does not, then, given any number Δ, however large, φ ( n ) > Δ for all sufficiently large values of n , and so φ ( n ) + . If on the other hand R exists, the classes L and R form a section of the real numbers in the sense of § 17 . Let a be the number corresponding to the section, and let be any positive number. Then φ ( n ) < a + for all values of n , and so, since is arbitrary, φ ( n ) 5 a . On the other hand φ ( n ) > a - for some value of n , and so for all sufficiently large values. Thus a - < φ ( n ) 5 a for all sufficiently large values of n ; i.e. φ ( n ) a. It should be observed that in general φ ( n ) < a for all values of n ; for if φ ( n ) is equal to a for any value of n it must be equal to a for all greater values of n . Thus φ ( n ) can never be equal to a except in the case in which the values of φ ( n ) are ultimately all the same. If this is so, a is the largest member of L ; otherwise L has no largest member. Cor 1. If φ ( n ) increases steadily with n , then it will tend to a limit or to + according as it is or is not possible to find a number K such that φ ( n ) < K for all values of n . We shall find this corollary exceedingly useful later on. Cor 2. If φ ( n ) increases steadily with n , and φ ( n ) < K for all values of n , then φ ( n ) tends to a limit and this limit is less than or equal to K . It should be noticed that the limit may be equal to K : if e.g. φ ( n ) = 3 - (1 /n ), then every value of φ ( n ) is less than 3, but the limit is equal to 3.
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[IV : 71] POSITIVE INTEGRAL VARIABLE 159 Cor 3. If φ ( n ) increases steadily with n , and tends to a limit, then φ ( n ) 5 lim φ ( n ) for all values of n . The reader should write out for himself the corresponding theorems and corollaries for the case in which φ ( n ) decreases as n increases. 70. The great importance of these theorems lies in the fact that they give us (what we have so far been without) a means of deciding, in a great many cases, whether a given function of n does or does not tend to a limit as n → ∞ , without requiring us to be able to guess or otherwise infer beforehand what the limit is . If we know what the limit, if there is one, must be, we can use the test | φ ( n ) - l | < ( n = n 0 ) : as for example in the case of φ ( n ) = 1 /n , where it is obvious that the limit can only be zero. But suppose we have to determine whether φ ( n ) = 1 + 1 n n tends to a limit. In this case it is not obvious what the limit, if there is
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