Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/43

Before proving these theorems it is necessary to establish the following properties of infinitesimals.

(1) The sum of a finite number of infinitesimals is an infinitesimal. To prove this we must show that the numerical value of this sum can be made less than any small positive quantity (as $\scriptstyle {\epsilon }$) that may be assigned (§15). That this is possible is evident, for, the limit of each infinitesimal being zero, each one can be made numerically less than $\scriptstyle {\frac {\epsilon }{n}}$ ($\scriptstyle {n}$ being the number of infinitesimals), and therefore their sum can be made numerically less than $\scriptstyle {\epsilon }$.

(2) The product of a constant $\scriptstyle {c}$ and an infinitesimal is an infinitesimal. For the numerical value of the product can always be made less than any small positive quantity (as $\scriptstyle {\epsilon }$) by making the numerical value of the infinitesimal less than $\scriptstyle {\frac {\epsilon }{c}}$.

(3) The product of any finite number of infinitesimals is an infinitesimal. For the numerical value of the product may be made less than any small positive quantity that can be assigned. If the given product contains $\scriptstyle {n}$ factors, then since each infinitesimal may be assumed less than the $\scriptstyle {n}$th root of $\scriptstyle {\epsilon }$, the product can be made less than $\scriptstyle {\epsilon }$ itself.

(4) If $\scriptstyle {v}$ is a variable which approaches a limit $\scriptstyle {l}$ different from zero, then the quotient of an infinitesimal by $\scriptstyle {v}$ is also an infinitesimal. For if limit $\scriptstyle {v=l}$, and $\scriptstyle {k}$ is any number numerically less than $\scriptstyle {l}$, then, by definition of a limit, $\scriptstyle {v}$ will ultimately become and remain numerically greater than $\scriptstyle {k}$. Hence the quotient $\scriptstyle {\frac {\epsilon }{v}}$, where $\scriptstyle {\epsilon }$ is an infinitesimal, will ultimately become and remain numerically less than $\scriptstyle {\frac {\epsilon }{k}}$, and is therefore by (2) an infinitesimal.

Proof of Theorem I. Let $\scriptstyle {v_{1},v_{2},v_{3},\cdots }$ be the variables, and $\scriptstyle {l_{1},l_{2},l_{3},\cdots }$ their respective limits. We may then write ${\begin{array}{c}\scriptstyle {v_{1}-l_{1}=\epsilon _{1},}\\\scriptstyle {v_{2}-l_{2}=\epsilon _{2},}\\\scriptstyle {v_{3}-l_{3}=\epsilon _{3},}\\\scriptstyle {.~.~.~.~.}\end{array}}$where $\scriptstyle {\epsilon _{1},\epsilon _{2},\epsilon _{3},\cdots }$ are infinitesimals (i.e. variables having zero for a limit). Adding