A question about consistency

Question

These Definitions come from Analysis I textbook of Tao, My question is: what is mean of 'Definition 6.1.2 is consistent with Definition 4.3.4',and why is it so clear that they are consistent. How do we verify this 'consistency'? Thank you!

Answer 1

In this case, consistency means the two definitions agree in the case where both are applicable. I.e. if we take $\varepsilon >0$ rational as well as $x,y$ rational, we could use Def.4.3.4 to see that $x$ and $y$ are $\varepsilon$-close as rational numbers if $d(x,y)\leq \varepsilon$. On the other hand, every rational number is also a real number, so we could as well use Def.6.1.2 to see they are $\varepsilon$-close if $d(x,y)\leq \varepsilon$. Obviously the two conditions are the same (as $d(x,y)$ on rational numbers is the same as $d(x,y)$ on real numbers for $x,y$ rational [hopefully; otherwise the definitions would not be consistent]), so it does not cause any trouble if we call them $\varepsilon$-close in the sense of Def. 4.3.4 or Def. 6.1.2.

We defined something in a special case and now use the same term for a larger family of things, but we have to ensure our previous definition means the same thing on the subset used earlier, otherwise we have a problem.

I guess the main point is to emphasize that the $d(x,y)$ on $\Bbb R$ gives you the same values as $d(x,y)$ on $\Bbb Q$ if $x,y\in\Bbb Q$.

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Maybe an example of inconsistency:

Def 1: We call a number $x\in \Bbb N$ a "fun number" if it is divisible by 2.

Def 2: We call a number $x\in \Bbb Z$ a "fun number" if it is divisible by 4.

Well now the number $2\in \Bbb N\subseteq \Bbb Z$ is both a "fun number" (by Def. 1) and not (by Def. 2). This is bad.