The results in Chapter were particular statements. A particular statement either asserts that a property, or predicate, is true of a subjects (e.g “seven is prime”; “Socrates is mortal”), or asserts some relation between two or more things (e.g. $\sqrt[8]{8!}\lt \sqrt[9]{9!;}1/1,000\lt 1/1,000,000$). In any science, but especially in mathematics, we are interested in generalising from particular facts. When we saw that,
$$\frac{1}{1,000}-\frac{1}{1,001}\lt \frac{1}{1,000,000}$$
we realised that something similar would have happened whenever we subtracted the reciprocals of two consecutive whole numbers. A similar proof would have shown, for example, that,
$$\frac{1}{203}-\frac{1}{204}\lt\frac{1}{(203)^{2}}$$
The particular result that,
$\frac{1}{1,000}-\frac{1}{1,001}\lt \frac{1}{1,000,000}$
is an instance of the general pattern: for all whole numbers n,
$$\frac{1}{n}-\frac{1}{n+1}\lt \frac{1}{n^{2}}$$
This is an example of an “all” statement, or a “universal generalisation” in the jargon of logic. (Unfortunately, logic is a subject especoally given to producing jargon; a precise technical language is sometimes needed, but should not be overdone.)
Some equivalent ways to express the same result in english are:
For every whole number n,
$$\frac{1}{n}-\frac{1}{n+1}\lt \frac{1}{n^{2}}$$
For any whole number n,