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Intersection of closed sets

In the metric space \left( \mathbb{Q} , \left| \cdot \right| \right) , find a decreasing sequence \left \{ A_n \right \}_{n \in \mathbb{N}} of non closed subsets of \mathbb{Q} such that \mathrm{diam} \left ( A_n \right ) \xrightarrow{n \rightarrow +\infty} 0 and \bigcap \limits_{n=1}^{\infty} A_n =\varnothing.

Solution

Let us pick A_{n}=\mathbb{Q}\cap \left[\sqrt{2}-\frac{1}{n},\sqrt{2}+\frac{1}{n} \right]. Indeed,

    \[\mathrm{diam}(A_n)=\left|\sqrt{2}+\frac{1}{n}-\left(\sqrt{2}-\frac{1}{n}\right) \right|=\frac{2}{n}\xrightarrow{n \rightarrow +\infty} 0\]

and

    \begin{align*} \bigcap_{n=1}^{\infty} A_n &= \bigcap_{n=1}^{\infty} \left (\mathbb{Q}\cap \left[\sqrt{2}-\frac{1}{n},\sqrt{2}+\frac{1}{n} \right] \right ) \\ &=\left ( \bigcap_{n=1}^{\infty} \left [ \sqrt{2} - \frac{1}{n} , \sqrt{2} + \frac{1}{n} \right ] \right ) \cap \mathbb{Q} \\ &= \left \{ \sqrt{2} \right \} \cap \mathbb{Q}\\ &= \varnothing \end{align*}

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