Showing that the square root is monotone I've shown the existence of unique square roots of all positive rational numbers, so now I want to prove that the square root is monotone: 
$0<a<b$ if and only if $\sqrt{a} < \sqrt{b}$

We know that if $p, q, r, s$ are positive and $p < q, r < s$ then $pr < qs$. Let $\sqrt{a} < \sqrt{b}$ using $p = r = \sqrt{a}, q = s = \sqrt{b}$ we get $a < b$. 
Let $a < b$. Clearly we can't have $\sqrt{a} = \sqrt{b}$ as this will mean $a = b$ (by squaring). Also we can't have $\sqrt{a} > \sqrt{b}$ as by previous part it would mean $a > b$. Hence we must have $\sqrt{a} < \sqrt{b}$.
