Just wondering if my answers of the following 2 part question are correct. If not i would much appreciate a layman explanation, where i stuffed up.
Cheers
Find lim sup $A_n$ and lim inf $A_n$ of the following sequence of sets:
$A_n = \begin{cases} x &0 \le x < 1 &\text{if n is odd} \\ x &1 \le x \le 2 &\text{if n is even} \end{cases}$
Answer Attempt lim sup $A_n = \{0, 2\}$ and lim inf $A_n = \emptyset$
$E_n = \begin{cases} x &-n \le x \le 0 &\text{if n is odd} \\ x &\frac{1}{n} \le x \le n &\text{if n is even} \end{cases}$
Answer Attempt lim sup $A_n = \{-\infty, 0\} \cup \{1/2, \infty\}$ and lim inf $A_n = \emptyset$
I am skeptical if i am doing this right as both of my lim inf were null sets. One would imagine out of only 2 exercises on lim sup and lim inf of sets at the end of the chapter one would have been different.
Any help would be much appreciated.
Definitions from the book. For a sequence of subsets $A_n$ of a set $X$, the $\limsup A_n= \cap_{N=1}^\infty ( \cup_{n\ge N} A_n )$ and $\liminf A_n = \cup_{N=1}^\infty (\cap_{n \ge N} A_n)$.