Baby Rudin 1.1, the need of real numbers The investigation:

Why is the claim that "A contains no largest number and B contains no smallest" tantamount to saying that there's a gap at $ \sqrt{2}$ (in terms of rationals)? Rudin specifically restricts $p^2 < 2 $ and $p^2 > 2$, so how can that be used to justify the fact that there's no rational $p = \sqrt{2}$. I get the first part of the example, and I understand why there's no largest and smallest rational number in those sets, but there would also be no largest or smallest real in analogous real number sets, isn't that true?
 A: There are indeed order-gaps in $\mathbb Q.$ We have $\mathbb Q=\{q\in \mathbb Q:q^2<2\}\cup \{q\in \mathbb Q: q^2>2\}.$ 
$$ \text {Therefore }\quad \mathbb Q=A\cup B$$ $$\text {where }\quad  A=\{q\in\mathbb Q:q\leq 0\lor (q>0\land q^2<2)\}$$ $$\text {and }\quad B=\{q\in \mathbb Q: 0<q\land q^2>2\}.$$ Every member of $A$ is less than every member of $B$ so every $a\in A$ is a lower bound for $B$ and every $b\in B$ is an upper bound for A. 
The point is that, because $A$ has no largest member and $B$ has no least member, therefore no member of $\mathbb Q$ is a $lub$ for $A,$ and no member of $\mathbb Q$ is a $glb$ for $B.$
So a sequence of members of $\mathbb Q,$ that heuristically "should" converge to a number, may fail to converge to a member of $\mathbb Q.$
BTW. Another way to show that $B$ has no $\min$ and that  $A$ has no $\max$ is to use a method written about by Hero (Heron) of Alexandria circa 100 A.D. for finding approximate square roots:
(i). For $y>0$ and $x>0$ with $x^2>y,$ let $x'=\frac {1}{2}(x+\frac {y}{x}).$ Then $0<x'<x$ and $x'^2>y.$ In particular, with $y=2$ and $x\in B$ we have $x>x'\in B.$. 
(ii). For $y>0$ and $z>0$ with $z^2<y,$ let $x=\frac {y}{z}$ and let $x'$ be as in (i) above, and let $z^*=\frac {y}{x'}.$ Then $0<z<z^*$ and $(z^*)^2<y.$ In particular with $y=2$ and $0<z\in A$ we have $z<z^*\in A.$
BTW. For $x_1>0$ and $y>0,$ let $x_{n+1}=\frac {1}{2}(x_n+\frac {y}{x_n}).$  The sequence $(x_n)_{n\in \mathbb N}$ converges (in $\mathbb R$) rapidly to $\sqrt y\;,$ which is what Heron wrote about.
