用一列区间覆盖 $\mathbb{Q}$ 总长可任意小

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How We Got From There To Here: A Story of Real Analysis
Corollary 12.1.10 给定 $\varepsilon>0$. 存在 $\mathbb{R}$上的区间 $I_n=\left[a_n, b_n\right]$ 使得
$$
\mathbb{Q} \subset \bigcup_{n=1}^{\infty} I_n
$$
且
$$
\sum_{n=1}^{\infty}\left(b_n-a_n\right)<\varepsilon
$$

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设$\mathbb Q = \{a_1, a_2, a_3,...\}$
$I_n=(a_n-\varepsilon/2^n/2, a_n+\varepsilon/2^n/2)$
$I_n$的总长为$∑_{n=1}^\infty\varepsilon/2^n=\varepsilon$, 并且覆盖$\mathbb Q$.