Following up from the discussion here: Liminf and Limsup of a sequence of sets
I wanted to confirm my understanding of these concepts with another example. Suppose we have: $a_n>0$, $b_n >1$ and
$$ \lim_{n\rightarrow \infty}a_n =0,\ \lim_{n\rightarrow \infty}b_n =1$$
and
$$A_n=\left \{ x:a_n \leq x I view this as both $a_n$ and $b_n$ being decreasing sequences of real numbers. To help with enumeration, I have defined $a_n=\frac{1}{n}, b_n=1+\frac{1}{n}$. Then $A_1= \left \{ x: 1 \leq x < 2 \right \} = [1,2),A_2= \left \{ x: 1/2 \leq x < 3/2 \right \} = [1/2,3/2)$ and so on. To me, $A_n$ appears to be a sequence of half-open intervals of the form [a,b). I reason that this means the liminf and limsup (and hence the limit) are (0,1]. Is this line of thinking right?