Two examples

[Grigorios Kostakos]

1. Give an example of a descending sequence $(F_n)_{n\in\mathbb{N}}$ of non-empty closed subsets of metric space $(\mathbb{R}, |\cdot|)$, such that $\bigcap_{n=1}^{\infty}F_n=\varnothing$.
2. Give an example of a descending sequence $(F_n)_{n\in\mathbb{N}}$ of non-empty closed subsets of metric space $(\mathbb{Q}, |\cdot|)$, such that ${\rm{diam}}(F_n)\xrightarrow{n\to+\infty} 0$ and $\bigcap_{n=1}^{\infty}F_n=\varnothing$.