83. The general principle of convergence for a bounded function.
The results of the preceding sections enable us to formulate a very important necessary and sufficient condition that a bounded function \(\phi(n)\) should tend to a limit, a condition usually referred to as the general principle of convergence to a limit.
Theorem 1. The necessary and sufficient condition that a bounded function \(\phi(n)\) should tend to a limit is that, when any positive number \(\epsilon\) is given, it should be possible to find a number \(n_{0}(\epsilon)\) such that \[|\phi(n_{2}) – \phi(n_{1})| < \epsilon\] for all values of \(n_{1}\) and \(n_{2}\) such that \(n_{2} > n_{1} \geq n_{0}(\epsilon)\).
In the first place, the condition is necessary. For if \(\phi(n) \to l\) then we can find \(n_{0}\) so that \[l – \tfrac{1}{2}\\epsilon < \phi(n) < l + \tfrac{1}{2}\epsilon\] when \(n \geq n_{0}\), and so \[\begin{equation*} |\phi(n_{2}) – \phi(n_{1})| < \epsilon \tag{1} \end{equation*}\] when \(n_{1} \geq n_{0}\) and \(n_{2} \geq n_{0}\).
In the second place, the condition is sufficient. In order to prove this we have only to show that it involves \(\lambda = \Lambda\). But if \(\lambda < \Lambda\) then there are, however small \(\epsilon\) may be, infinitely many values of \(n\) such that \(\phi(n) < \lambda + \epsilon\) and infinitely many such that \(\phi(n) > \Lambda – \epsilon\); and therefore we can find values of \(n_{1}\) and \(n_{2}\), each greater than any assigned number \(n_{0}\), and such that \[\phi(n_{2}) – \phi(n_{1}) > \Lambda – \lambda – 2\epsilon,\] which is greater than \(\frac{1}{2}(\Lambda – \lambda)\) if \(\epsilon\) is small enough. This plainly contradicts the inequality (1). Hence \(\lambda = \Lambda\), and so \(\phi(n)\) tends to a limit.
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 \(\phi(n)\) tends to a limit \(l\) then it is certainly bounded; for all but a finite number of its values are less than \(l + \epsilon\) and greater than \(l – \epsilon\).
In the second place, if the condition of Theorem 1 is satisfied, we have \[|\phi(n_{2}) – \phi(n_{1})| < \epsilon\] whenever \(n_{1} \geq n_{0}\) and \(n_{2} \geq n_{0}\). Let us choose some particular value \(n_{1}\) greater than \(n_{0}\). Then \[\phi(n_{1}) – \epsilon < \phi(n_{2}) < \phi(n_{1}) + \epsilon\] when \(n_{2} \geq n_{0}\). Hence \(\phi(n)\) 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 \(\phi(n)\) 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.1 We will only remark that, if we suppose that \[\phi(n) = s_{n} = u_{1} + u_{2} + \dots + u_{n},\] 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 \(u_{1} + u_{2} + \dots\) is that, given any positive number \(\epsilon\), it should be possible to find \(n_{0}\) so that \[|u_{n_{1}+1} + u_{n_{1}+2} + \dots + u_{n_{2}}| < \epsilon\] for all values of \(n_{1}\) and \(n_{2}\) such that \(n_{2} > n_{1} \geq n_{0}\).
- A few proofs given in Ch. VIII can be simplified by the use of the principle.↩︎
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