Prove the existence of the square root of 2

**dknuth** · October 16th 2011, 10:18 AM

I'm currently in Calculus I Honors at my university and so far, after 1.5 months of instruction, we've yet to touch calculus... But in the mean time, my professor has gone deep into number theory, and we spent the first 2 weeks proving the existence of real numbers using a Cauchy Sequence. For our first exam, he prompted us with the extra-credit:

Prove the existence of the square root of 2 using a Cauchy Sequence of rationals. (Also, feel free to re-explain Cauchy Sequences to me........)

THANK YOU SO MUCH!

**Deveno** · October 17th 2011, 01:50 AM

consider the following sequence:

$x_0 = 1$

$x_{n+1} = \frac{1}{2}\left(x_n+\frac{2}{x_n}\right)$

is this sequence cauchy? a cauchy sequence is one for which:

$\forall \epsilon > 0,\ \exists N \in \mathbb{Z}^+$ such that

$|x_m - x_n| < \epsilon,\ \forall \ m,n > N$

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