The length of a gap between the rationals

Question

In Terence Tao's book Analysis I, he says

there are still an infinite number of “gaps” or “holes” between the rationals, although this denseness property does ensure that these holes are in some sense infinitely small.

I think a gap between the rationals should have zero length.

Supposing $A_{1}=\{a\in {\mathbb {Q}}:a^{2}<2{\text{ or }}a<0\}, {\displaystyle A_{2}=\{a\in \mathbb {Q} :a^{2}>2{\text{ and }}a>0\}} $, I define the "length" of the gap between $A_{1}$ and $A_{2}$ to be the greatest lower bound of $A =\left\{ a \middle| a = a_{2} - a_{1}, {\ a}_{1} \in A_{1}{,a}_{2} \in A_{2} \right\}$, so how to prove the greatest lower bound is $0$ ? especially using the density property of rational numbers to prove it?

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Maybe I have a lack of understanding in the density property of rational numbers , so I am unable to give a proof to my question here.

Answer 1

It's a good question! Denseness alone is not enough. Consider the set $P=\mathbb Q\setminus[1,2]$. This $P$ is also dense as an ordered set. (But it is not dense as a subset of $\mathbb R$.)

Now let $B_{1}=\{a\in P:a^{2}<2{\text{ or }}a<0\}, {\displaystyle B_{2}=\{a\in P :a^{2}>2{\text{ and }}a>0\}} $, and $B =\left\{ a \middle| a = a_{2} - a_{1}, {\ a}_{1} \in B_{1}{,a}_{2} \in B_{2} \right\}$. This is a recapitulation of your $A$; the only difference is that $\mathbb Q$ is replaced by $P$. You can see that $\inf B=1$!

So you cannot prove that $\inf A=0$ merely from the knowledge that $\mathbb Q$ is dense. You need something else: for example, that $\mathbb Q$ is closed under averages, which is after all how Proposition 4.4.3 is proved.

Answer 2

It is possible to prove the greatest lower bound of $A =\left\{ a \middle| a = a_{2} - a_{1}, {\ a}_{1} \in A_{1}{,a}_{2} \in A_{2} \right\}$ is 0 in the rational number system， since if any other positive number $b$ is the greatest lower bound, we could always find a positive number $c=a_{2} - a_{1}<b$ . I think what is important here is that the question won't make much sense unless we prove the greatest lower bound of $A$ is 0 in a continuous number system ($\mathbb R$). We may use Archimedean property of $\mathbb R$ to prove the conclusion .