What is the definition of limit of a sequence of sets? Here's an example.
Let $s\in \mathbb{Z}^+$ and $q=2s+1$.
Define $T_{s,i}(x)=\frac{x}{q} + \frac{2i}{q}$ where $x\in\mathbb{R}$ and $i=0,1,\ldots,s$.
Define recursively $I_{s,0} = [0,1]$ and $I_{s,n+1} =\bigcup_{i=0}^s T_{s,i} (I_{s,n})$
Let $\Gamma (s) = \bigcap_{n\in\omega} I_{s,n}$
The text i'm reading says $\lim_{s\to \infty} \Gamma (s)$ does not exist but $\lim\inf_{s\to \infty} \Gamma (s)$ exists.
I have no idea what this limit refers to.. 

Given a family of sets $(X_i)_{i\in\Bbb N}$, you can define its upper and lower limits as follows :
$$\mathrm{lim~inf}~X_i=\bigcup_{n=0}^{\infty}\bigcap_{k=n}^{\infty}X_k$$ and $$\mathrm{lim~sup}~X_i=\bigcap_{n=0}^{\infty}\bigcup_{k=n}^{\infty}X_k.$$ The first set is the set of all elements that appear in all $X_n$ provided $n$ is large enough, and the second is the set of all elements that never disappear, that is, for all $n$ there is $N\geq n$ such that $X_N$ contains it. Obviously
$$\mathrm{lim~inf}~X_i\subset \mathrm{lim~sup}~X_i$$
and I've seen some older probability texts define the sequence $(X_i)$ to have a limit whenever the two sets are equal. One then sets $$\lim X_i=\mathrm{liminf}~X_i= \mathrm{limsup}~X_i.$$
