Open cover rationals proper subset of R?

Solution 1:

Maybe this is silly, but for a rational $q<\sqrt 2$ take the open interval to be $(-\infty,\sqrt 2)$, and for a rational $q>\sqrt 2$ take the open interval to be $(\sqrt 2,\infty)$. This would furnish a counterexample.

Solution 2:

If you enumerate the rationals as a sequence $x_1, x_2, \dots$, you can then take a sequence of open intervals $(x_1-\delta, x_1+\delta), (x_2-\delta/2, x_2+\delta/2), (x_3-\delta/4, x_3+\delta/4), \dots$ which gives an open cover for $\mathbb{Q}$ of total length $4\delta$, which can be made as small as you wish, by choosing $\delta$ sufficiently small.