Let $A_n$ be the interior of the circle with center at $( (-1)^n/n,0) )$ and radius $1$.
In other words, $A_n$ = { $ (x,y) | (x -(-1)^n/n )^2 + (y -0)^2 < 1$}.
What is the $\limsup_n A_n$ and what is the $\liminf_nA_n$?
My approach to a similar problem was to look at the limit of the sets and argue from there. It went something like as $\limsup_nA_n$ contains infinitely recurring points of the $A_n$, the limit of the circles will be contained in infinitely many of the $A_n$, so $\limsup_nA_n \supseteq \{ (x,y) | x^2 + y^2 < 1 \} $. Further, as any $x\in \limsup_nA_n$ lies in infinitely many $A_n$, $\limsup_nA_n \subseteq \lim_nA_n$, so the two sets are equal.
However, I'm a bit stuck from there. The book says
$\liminf_nA_n = \{ (x,y)| x^2 + y^2 \leq \} - \{ (0,1) , (0,-1) \}.$
Yet, I have no idea how they arrived at this.
Some help would be greatly appreciated, please and thank you.
You probably mean $$\liminf_nA_n = \{ (x,y)\in\mathbb R^2\mid x^2 + y^2 \lt1 \},$$ and $$\limsup_nA_n= \{ (x,y)\in\mathbb R^2\mid x^2 + y^2 \leqslant1 \} \setminus\{ (0,1) , (0,-1) \}.$$ To show the liminf, one must determine the points $(x,y)$ that are in $A_n$ for every $n$ large enough. To show the limsup, one must determine the points $(x,y)$ that are in $A_n$ for infinitely many $n$. There is no real difficulty to do so, especially when the answers are given.
Note that the sets $A_n$ have no limit hence any approach requiring "to look at the limit of the sets and argue from there" is doomed.