Definition of a Sequence

The title of Section 2 in book [1] is Sequences of rational numbers. At the beginning of this section, a definition is provided for a sequence, and I quote it here.

Definition. If, by means of any suitable process of construction, we can form successively a first, a second, a third, … (rational) number and if to every positive integer $ n$ one and only one well-defined (rational) number $ x_n$ thus corresponds, then the numbers $ x_1, x_2, x_3, …, x_n, …$ (in this order, corresponding to the natural order of the integers $ 1,2,3,…,n,…$) are said to form a sequence. We denote it for brevity by $ (x_n)$ or $ (x_1, x_2, …)$.

A sequence can be useful for understanding the value of $ \sqrt{2}$ because a sequence can consist of rational numbers that, when squared, become closer to $ 2$ as $ n$ increases, as mentioned in reference [1]. The author of [1] claims that the following definition is very important, so I will copy it here as well. At the moment, I do not know why this “null sequence” is important.

Definition. A sequence will be called a null sequence if it possesses the following property: given any arbitrary positive (rational) number $ \epsilon$, the inequality $ |x_n| < \epsilon$ is satisfied by all the terms, with at most a finite number of exceptions. In other words: an arbitrary positive number $ \epsilon$ being chosen, it is always possible to designate a term $ x_m$ of the sequence, beyond which the terms are less than $ \epsilon$ in absolute value. Or a number $ n_0$ can always be found, such that $ |x_n| < \epsilon$ for every $ n >n_0$.

One pictorial representation of a null sequence is:

References