What is a good way to show that $[0,1]$ is not complete in $\mathbb{Q}$

Question

To show a set is not complete, the best way is always produce a Cauchy sequence that does not converge in the set.

I wish to show $[0,1]$ is not complete in $\mathbb{Q}$

I am a little stucking procuring a cauchy sequence that does not converge for this set.

Some ideas:

• Decimal expansion: but $e$ and $\pi$ are outside of $[0,1]$...

• Some well known examples: take $p_n = \frac{1}{1+n}, n \in \mathbb{Q}$, actually this one is not very good. Yup.

Any other ideas?

Answer 1

If you know $e$ or $\pi$ is irrational, you can use $e-2$ or $\pi-3$

Answer 2

Let $x_0 = {3 \over 4}$ and $x_{n+1} = {2 x_n^2+1 \over 4x_n}$. It is straightforward to see that $x_n \in \mathbb{Q}$ for all $n$. This is the Newton iteration for solving $x^2 = {1 \over 2}$.

A little work shows that $x_n$ is Cauchy, but does not converge (in $\mathbb{Q}$, it converges rather quickly to ${1 \over \sqrt{2}}$).