I remember I saw this question somewhere in Lang's undergraduate real analysis.
Given any real number $\ge0$, show that it has a square root.
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2$\begingroup$ You can give an iteration sequence for it and show that it is Cauchy. Then use the completeness of $\mathbb R$. $\endgroup$– AlexRCommented Sep 24, 2013 at 8:21
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2 Answers
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It depends on how you define a real number. If you use Dedekind cuts, then you should show that the set $\{ x \in \mathbb{Q}^+: x^2<2\}$ is a Dedekind cut. If you use Cauchy sequences to define a real number, you can prove that the the sequence that is obtained from Newton's method is Cauchy:
$$ a_{n+1} = \frac{a_n}{2} + \frac{1}{a_n}$$
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This means to show that for $a\geq 0$, the Polynomial $x^2-a$ has at least one real root. Chose $x_0 := 1$ and use Newton's method with $$x_{n+1} = x_n - \frac{x_n^2 - a}{2x_n}$$ Then since $(x_n)$ is cauchy and $a \geq 0$ is a fixpoint of the iteration $$\lim_{n\to\infty} x_n^2 = a$$ $x_n \to \sqrt{a}$ and since $\mathbb R$ is complete, $\sqrt{a} \in \mathbb R$.
