Counter examples for closed and bounded sets

Question

(1) Given an example of sets $A_1\subseteq A_2 \subseteq\dots$ in $\mathbb{R}$ such that each $A_n$ is bounded and $$\bigcap ^{\infty}_{n=1}A_n= \emptyset.$$

(2) Given an example of sets $A_1\subseteq A_2 \subseteq\dots $ in $\mathbb{R}$ such that each $A_n$ is closed and $$\bigcap ^{\infty}_{n=1}A_n= \emptyset.$$

My attempt:

For (2) $A_n=[n,\infty)$

(1) $\left\{\left(0,\frac{1}{n}\right) \right\}^\infty _{n=1}$

Am I correct?

Answer 1

Nope this is incorrect. In fact with $A_1\subseteq A_2 \subseteq..... \subset \mathbb{R}$ we have $$\bigcap ^{\infty}_{n=1}A_n=A_1$$ therefore $$A_1=\emptyset.$$