Homework 3.

1. For $n\ge 1$, $a_{3n-2}=\frac{n}{n+1},a_{3n-1}=\frac{1}{n},a_{3n}=10$. What are the limitpoints of the sequence ${\{a_n\}}_{n=1}^{\mathrm{\infty }}$?

   [2 points]

2. What is the limitpoint set of the sequence $1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,\mathrm{\dots }.$

   [2 points]

3. Let ${\{x_n\}}_{n=1}^{\mathrm{\infty }},{\{y_n\}}_{n=1}^{\mathrm{\infty }}$ be bounded sequences. Show that

   $$sup{\{x_n\}}_{n=1}^{\mathrm{\infty }}+sup{\{y_n\}}_{n=1}^{\mathrm{\infty }}\ge sup{\{x_n+y_n\}}_{n=1}^{\mathrm{\infty }}.$$

   [3 points]

4. Let $F\in ℝ$ be a closed set. Let $y\in ℝ$. Let $F_1$ be the set of points $x$ in $F$ such that $y+x\le 2$. Show that $F_1$ is a closed set as well.

   [3 points]