Existence of square root

Solution 1:

If you are using Completeness Axioms (like existence of least upper bound (LUB) of a set bounded above) then (If I recollect correctly) the "standard" approach is to consider (assuming $r \in (0,1)$)

$$ S = \{ x : x^2 \lt r, \ x \in [0,1]\}$$

and prove that the LUB (or supremum) $l$ of $S$ satisfies $l^2 = r$ by showing

(Hint)

$l^2 \le r$ and that if $l^2 \lt r$, then there is some $s \gt l$ such that $s^2 \lt r$

(Spoiler)

Given any $\epsilon \gt 0$, there is an $x \in S$ such that $x \gt l - \epsilon$.
Squaring and using the fact that $l \lt 1$ (Why?) leads to $r \gt l^2 - 2 \epsilon$
This implies that $l^2 \le r$ (Why?)

If $a = r - l^2 \gt 0$, then we have that by choosing $\epsilon = \frac{a}{3}$ that $\epsilon \lt 1$ (Why?) and $a \gt 2\epsilon + \epsilon^2 \gt 2 \epsilon l + \epsilon^2$ and thus $r \gt (l + \epsilon)^2 = s^2$ with $s \gt l$.
We can also show that $s \lt 1$ and thus $s \in S$ contradicting that $l = \sup S$

Note that the proof above gives rise to the sequence

$x_{n+1} = x_n + \dfrac{r - x_n^2}{3}$ which is quite similar to Henry's sequence.

From what I could see from Google books, the book you are using does use that axiom.

Solution 2:

(i) Let $s_0 = r$ and $s_{n+1} = s_n + \dfrac{r-s_n^2}{2}$

(ii) If $a \gt 1$, let $r=\dfrac{1}{a}$ and use (i)