# Counter examples for closed and bounded sets

(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?

• You have the containment sign backwards in statement (1). Other than that, it looks good. – saulspatz Mar 8 '18 at 3:18
• Your answers do not satisfy $A_k \subseteq A_{k+1}$. The reverse containment is satisfied. Indeed, the stated requirements cannot be satisfied unless $A_1 = \emptyset$, for both problems. – Bungo Mar 8 '18 at 3:18
• @Bungo.. you mean that not satisffies increasing property right? – Inverse Problem Mar 8 '18 at 3:20
• Correct, your sequences are both decreasing, not increasing. Are you sure the problem statements should not request $A_1 \supseteq A_2 \supseteq \cdots$? – Bungo Mar 8 '18 at 3:21
• @Bungo..please can you give me examples – Inverse Problem Mar 8 '18 at 3:22

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 $$A_1=\emptyset.$$