Workshop Exercises 3.

1. Give an example of a NON-CONVERGENT sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ such that its limitpoint set contains only the number $2017$.

2. Give an example of an UNBOUNDED sequence ${\{y_n\}}_{n=1}^{\mathrm{\infty }}$ such that its limitpoint set contains only the numbers $\{1,2\}$. the elements $1$ and $2$.

3. Show that the set $[1,2)\cup (2,3]$ is not closed.

4. Give an example of a sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ such that their limitpoints are exactly the prime numbers.

5. Let $F_1,F_2,\mathrm{\dots },$ be closed sets in $ℝ$ . Is it true that ${\cup }_{n=1}^{\mathrm{\infty }}{F}_n$ is always closed?

6. Let $F$ be a closed set in $ℝ$. Let $y\notin F$. Show that there exists some $\epsilon >0$ such that the intersection of $F$ and the open interval $(y-\epsilon ,y+\epsilon )$ is empty.

7. Let $F$ be a closed set $\in ℝ$. Let $y\notin F$. Show that ${inf}_{x\in F}|y-x|>0$. More challenging version: Show that there exists $a\in F$ such that $|a-y|={inf}_{x\in F}|y-x|$.

8. Even more challenging version. Let $F$ and $G$ be two disjoint sets in $ℝ$. Prove that ${inf}_{x\in F,y\in G}|x-y|>0$. Finally, show that there exists $a\in F$ and $b\in G$ such that $|a-b|={inf}_{x\in F,y\in G}|x-y|.$