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.