# Covering Rational Numbers with Arbitrarily Small Intervals

Take any arbitrary $\epsilon > 0$. There is a collection of intervals $\{I_n\}_{n \in \mathbb{N}}$ so that

\begin{aligned}\mathbb{Q} \subseteq \bigcup_{n = 1}^{\infty} I_n\end{aligned}

and the sum of their length never exceeds $\epsilon$.

To see this, for any $\epsilon > 0$, take any positive sequence $(a_n)$ converging to $\epsilon$. For example, $a_n = \dfrac{\epsilon}{2^n}$ will do. Since rational numbers are countably many, we can cover every rational number $r_n$ with open interval $I_n$ centered at $r_n$ with $\ell(I_n) = a_n$.

This is used to show that rational numbers, in the universe of real numbers, have measure zero, since we can control how small our $\epsilon$ is going to be.