The value of limsup and liminf of a sequence of a sets obtained by combining three sequences What is the limit superior of the following sequence of sets?
$\{X_n\}=\{\{1/2\},\{1/3\},\{1/4\},\{2/3\},\{1/3\},\{1/5\},\{3/4\},\{1/3\},\{1/6\}......\}(n\to∞)$
I.e., $X_1=\{1/2\}, X_2=\{1/3\}, X_3=\{1/4\}, \dots$ (I use the same notation as in the Wikipedia article linked below.)
I have taken the example from
Wikipedia article on limit superior where two sequences are combined. Here I have combined three sequences:


*

*the sequence $\{\frac{n}{n+1}\}$

*the constant sequence $\{\frac13\}$

*the sequence $\{\frac1n\}$


What's the value of
$\limsup{X_n},\liminf{X_n}$ and why? If it's possible to visualize?
From Wikipedia, I learn that they find subsequence first while if I don't know the number of subsequence, it seems hard to get answer. So I am confused and hope to visualize to understand it.
 A: From the post (and the comments) it seems that you are interested in Kuratowski limit superior.
There are obvious subsequences $x_{n_k}\in X_{n_k}$ which are convergent to $0$, $\frac13$ and $1$. If you check that for other real numbers there is no such subsequence, then you get that $$\limsup X_n=\{0,\frac13,1\}.$$ 

There is also another definition of limit superior of sets, namely 
$$\limsup X_n = \bigcap_{n=1}^\infty \bigcup_{m=n}^\infty X_m.$$
It can be considered the special case of Kuratowski limit superior in the case you use discrete topology (or discrete metric).
In this case you will get
$$\limsup X_n=\{\frac13\},$$
since $\frac13$ is the only number which appears in infinitely many $X_n$'s.
 Since I've mentioned two different notions of limit superior of sets, I'll add some links to other posts on this site, where these notions are discussed:


*

*Limit of sequence of sets - Some paradoxical facts

*Relations and differences between outer/inner limit and Kuratowski limsup/liminf

*$\limsup $ and $\liminf$ of a sequence of subsets relative to a topology
(Maybe also some other posts linked here.)
