A number of simple proofs of this result are given by

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* A number of simple proofs of this result are given by Hardy and Littlewood, “Some Problems of Diophantine Approximation”, Acta Mathematica , vol. xxxvii.
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[IV : 85] LIMITS OF FUNCTIONS OF A 184 84. Unbounded functions. So far we have restricted ourselves to bounded functions; but the ‘general principle of convergence’ is the same for unbounded as for bounded functions, and the words ‘ a bounded function ’ may be omitted from the enunciation of Theorem 1. In the first place, if φ ( n ) tends to a limit l then it is certainly bounded; for all but a finite number of its values are less than l + and greater than l - . In the second place, if the condition of Theorem 1 is satisfied, we have | φ ( n 2 ) - φ ( n 1 ) | < whenever n 1 = n 0 and n 2 = n 0 . Let us choose some particular value n 1 greater than n 0 . Then φ ( n 1 ) - < φ ( n 2 ) < φ ( n 1 ) + when n 2 = n 0 . Hence φ ( n ) is bounded; and so the second part of the proof of the last section applies also. The theoretical importance of the ‘general principle of convergence’ can hardly be overestimated. Like the theorems of § 69 , it gives us a means of deciding whether a function φ ( n ) tends to a limit or not, without requiring us to be able to tell beforehand what the limit, if it exists, must be; and it has not the limitations inevitable in theorems of such a special character as those of § 69 . But in elementary work it is generally possible to dispense with it, and to obtain all we want from these special theorems. And it will be found that, in spite of the importance of the principle, practically no applications are made of it in the chapters which follow. * We will only remark that, if we suppose that φ ( n ) = s n = u 1 + u 2 + · · · + u n , we obtain at once a necessary and sufficient condition for the convergence of an infinite series, viz: Theorem 2. The necessary and sufficient condition for the convergence of the series u 1 + u 2 + . . . is that, given any positive number , it should be possible to find n 0 so that | u n 1 +1 + u n 1 +2 + · · · + u n 2 | < for all values of n 1 and n 2 such that n 2 > n 1 = n 0 . * A few proofs given in Ch. VIII can be simplified by the use of the principle.
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[IV : 86] POSITIVE INTEGRAL VARIABLE 185 85. Limits of complex functions and series of complex terms. In this chapter we have, up to the present, concerned ourselves only with real functions of n and series all of whose terms are real. There is however no difficulty in extending our ideas and definitions to the case in which the functions or the terms of the series are complex. Suppose that φ ( n ) is complex and equal to ρ ( n ) + ( n ) , where ρ ( n ), σ ( n ) are real functions of n . Then if ρ ( n ) and σ ( n ) converge respectively to limits r and s as n → ∞ , we shall say that φ ( n ) converges to the limit l = r + is , and write lim φ ( n ) = l. Similarly, when u n is complex and equal to v n + iw n , we shall say that the series u 1 + u 2 + u 3 + . . .
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