19. Weierstrass’s Theorem

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.

We divide the points of the line $\mathrm{\Lambda }$ 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.