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.

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