About the fact that $\mathbb Q$ has gaps In my book there are two theorems:


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*$1.$ The number $\sqrt 2$ is irrational because if we put $\sqrt 2 = \frac pq$ for some integers  $p, q$ where $p, q$ have no common factors, then we get a contradiction.

*$2.$ The set $A = \mathbb Q \cap (0, \sqrt 2)$ has no largest number and $B = \mathbb Q \cap (\sqrt 2, \infty)$ has no smallest number.
Are the two theorems above saying the same thing or are they mutually exclusive? The reason I ask is because in a math book (AFAIK) the same statement is not usually reproven again after it's proven the first time without an explicit mention. So I was wonderig if there's some subtle difference between the two theorems above.
The technical part of the second theorem is not difficult, but I am having some uneasy time linking the fact that $A, B$ have no largest/smallest numbers with the fact that $\mathbb Q$ has gaps. Is the theorem saying that $\sqrt 2 \not \in A$ and $\sqrt 2 \not \in B$? But isn't that by very definition of $A, B$?
Thanks.

HINT.- If $s$ is the largest number of $A=\{x\in\mathbb Q: 0\lt x\lt\sqrt2\}$ then the non-empty interval $(s,\sqrt2)$ does not contain a rational (it is non-empty because we know $s$ is supossed rational). This is contrary to the known fact that $\mathbb Q$ is dense in $\mathbb R$. (There are other ways to prove this). Similarly with the other questions. The answer to your questions concerning the definition of $A$ and $B$ is negative.
It is known otherwise because $\mathbb Q$ is a set whose structure of order is not "well orderer"(contrary to the integers).
