What is the motivation behind Tao's properties of $\epsilon$-closeness? (elementary real analysis)

Question

In Analysis 1 by Terrence Tao, Tao introduces the concept of $\epsilon$-closeness, defined as follows

Let $\epsilon > 0$ be a rational number, and let $x, y$ be rational numbers. We say that $y$ is $\epsilon$-close to $x$ iff we have $|y - x| \leq \epsilon$.

He then goes on to give the following list of easy to prove properties of epsilon closeness.

1. If $x = y$, then $x$ is $\epsilon$-close to $y$ for every $\epsilon > 0$. Conversly, if $x$ is $\epsilon$-close to $y$ for every $\epsilon > 0$, then we have $x = y$

2. Let $\epsilon > 0$. If $x$ is $\epsilon$-close to $y$, then $y$ is $\epsilon$-close to $x$.

3. Let $\epsilon, \delta > 0$. If $x$ is $\epsilon$-close to $y$, and $y$ is $\delta$-close to $z$, then $x$ and $z$ are $(\epsilon + \delta)$-close.

4. Let $\epsilon > 0$. If $x$ and $y$ are $\epsilon$-close, they are also $\epsilon'$-close for every $\epsilon' > \epsilon$

5. Let $\epsilon > 0$. If $y$ and $z$ are both $\epsilon$-close to $x$, and $w$ is between $y$ and $z$ (i.e. $y \leq w \leq z$ or $z \leq w \leq y$), then $w$ is also $\epsilon$-close to $x$.

6. Let $\epsilon > 0$. If $x$ and $y$ are $\epsilon$-close, and $z$ is non-zero, then $xz$ and $yz$ are $\epsilon|z|$-close

7. Let $\epsilon, \delta > 0$. If $x$ and $y$ are $\epsilon$-close, and $z$ and $w$ are $\delta$-close, then $xz$ and $yw$ are $(\epsilon|z| + \delta|x| + \epsilon\delta)$ close.

Some of these properties can be easily motivated. $\epsilon$-closeness itself is easily motivated, as it is so essential to the formalization of calculus. 1-3 formalize the concept of $x$ and $y$ being $\epsilon$-close as "almost" equivalent. More precisely, they show that $\epsilon$-closeness is almost like to being an equivalence relation.

4 seems so basic and intuitive as to be worth proving.

However, 5-7 are less intuitive, especially 7.

Other lists of properties so far in this book have had clear motivation. For example, Tao has list of properties that are group, ring, and field axioms for $\mathbb{N}, \mathbb{Z},$ and $\mathbb{Q}$. Tao has properties showing that absolute value $|x - y|$ is a distance function, etc. But I am not aware of any underlying algebraic structure that these last 3 properties could be motivated with.

Here is my question:

Out of all the properties Tao could have put in this list, how did he know these would be sufficient to build the rest of the book off of? What made him choose these properties?

Answer 1

Probably from $(x-a)(y-b) =xy-xb-ya+ab $ so $|xy-(x-a)(y-b)| \le |xb|+|ya|+|ab| $.