Does there exist an enumeration of 
of 
such that
![\[\mathbb{R} \neq \bigcup_{n=1}^{\infty} \left( q_n - \frac{1}{n} , q_n + \frac{1}{n} \right)\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-346f8b47a4e0c8cb39d074476061f44c_l3.png "Rendered by QuickLaTeX.com")
Solution
Yes there exists. Let 
be enumerated in the following way:
- Enumerate
and let them be
. - Enumerate
![\mathbb{Q}\cap [0,1]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-89568fdec9f3406126ce6ac626bafc2f_l3.png "Rendered by QuickLaTeX.com")
and let them be 
where
is an increasing sequence which excludes all powers of 
.
Then the outer measure of
![\[\bigcup_{n=1}^{\infty} \left(q_n-\frac{1}{n}, q_n + \frac{1}{n} \right)\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-1e3e74a9427e3155660d28914b0c3dae_l3.png "Rendered by QuickLaTeX.com")
is finite.