The general theory of sets of points is of the utmost interest and importance in the higher branches of analysis; but it is for the most part too difficult to be included in a book such as this. There is however one fundamental theorem which is easily deduced from Dedekind’s Theorem and which we shall require later.
Theorem. If a set \(S\) contains infinitely many points, and is entirely situated in an interval \({[\alpha, \beta]}\), then at least one point of the interval is a point of accumulation of \(S\).
We divide the points of the line \(\Lambda\) into two classes in the following manner. The point \(P\) belongs to \(L\) if there are an infinity of points of \(S\) to the right of \(P\), and to \(R\) in the contrary case. Then it is evident that conditions (i) and (iii) of Dedekind’s Theorem are satisfied; and since \(\alpha\) belongs to \(L\) and \(\beta\) to \(R\), condition (ii) is satisfied also.
Hence there is a point \(\xi\) such that, however small be \(\delta\), \(\xi – \delta\) belongs to \(L\) and \(\xi + \delta\) to \(R\), so that the interval \({[\xi – \delta, \xi + \delta]}\) contains an infinity of points of \(S\). Hence \(\xi\) is a point of accumulation of \(S\).
This point may of course coincide with \(\alpha\) or \(\beta\), as for instance when \(\alpha = 0\), \(\beta = 1\), and \(S\) consists of the points \(1\), \(\frac{1}{2}\), \(\frac{1}{3}, \dots\). In this case \(0\) is the sole point of accumulation.
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