Do you have any simple example for nested bounded sets that have an empty intersection?

Question

Do you have any easy example for a family of nested BOUNDED sets $...\subset F_4\subset F_3\subset F_2\subset F_1$ such that: $$\bigcap_{n=1}^\infty F_n=\emptyset $$ in any metric space?

Answer 1

Take $F_n=\left(0,\frac1n\right)$ in $\Bbb R$.

Another easy example is to place the discrete metric,

$$d(m,n)=\begin{cases}0,&\text{if }m=n\\1,&\text{if }m\ne n\;,\end{cases}$$

on $\Bbb Z^+$, and let $F_n=\{k\in\Bbb Z^+:k\ge n\}$. The whole space is bounded, since the maximum distance between any two points is $1$, but $\bigcap_{n\in\Bbb Z^+}F_n=\varnothing$.

Answer 2

Take $F_n = \{\frac{1}{k} \}_{k \ge n}$.

(Or even simpler, take $F_n = \emptyset$.)

Answer 3

Let $\{q_n\mid n\in\Bbb N\}=\Bbb Q\cap[0,1]$. Let $F_n=\{q_k\mid k>n\}$. Then $\bigcap F_n=\varnothing$.