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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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You can give an iteration sequence for it and show that it is Cauchy. Then use the completeness of $\mathbb R$. – AlexR Sep 24 '13 at 8:21
up vote 17 down vote accepted

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, $\sqrt{a} \in \mathbb R$.

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