Theorem 2 (Principle of comparison)

Let $(x_i)_{i \in I}$ and $(y_i)_{i \in I}$ be two families of finite real numbers $\geq 0$, such that $x_i \leq y_i$ for all $i$. If $(y_i)$ is summable in $\mathbf{R}$ then so is $(x_i)$, and we have $\sum_i x_i \leq \sum_i y_i$; if in addition there is an index $x$ such that $x_x < y_x$, then $\sum_i x_i < \sum_i y_i$.

The hypothesis implies that,

for every finite subset $H$ of $I$ and the first part of the theorem follows from this; the inequality relating the sums follows from the principle of extension of inequalities ($§ 5$, no. 2, Theorem 1). If $x_x < y_x$, then

This theorem provides the most commonly used criterion for deciding whether or not a sequence $(x_n)$ of real numbers $\geq 0$ is summable in $\mathbf{R}$: we try to compare the given sequence with a simpler sequence $(y_n)$ of which we already know whether it is summable or not. If there exists a finite number $a > 0$ such that $x_n \leq a y_n$ for all $n$ from a certain point onwards, and if $(y_n)$ is summable, then $(x_n)$ is summable; if on the other hand there is a finite number $b > 0$ such that $x_n \geq b y_n$ for all $n$ from a certain point onwards, and if $(y_n)$ is not summable in $\mathbf{R}$, then $(x_n)$ is not summable in $\mathbf{R}$. We shall see in a later volume how such comparison sequences may be obtained in the cases which arise most frequently.