A question on $\liminf$ and $\limsup$ Let us take a sequence of functions $f_n(x)$. Then, when one writes $\sup_n f_n$, I understand what it means: supremum is equal to upper bound of the functions $f_n(x)$ at every $x$. Infimum is defined similarly. Then when one writes $\lim \sup f_n$, then I understand following: There are convergent subsequences of $f_n$, let us call them as $f_{n_k}$ and their limits as a set $E$. Then, $$\limsup f_n = \sup E$$
First question: Are these definitions right?
Second question: I do not understand the notion of convergent subsequences. What does it mean really? And why they are necessary at the first place, why they are important?
Thanks.

1 For any $ x $ there are $ n_{k(x)} $ such that 
\begin{equation}
\limsup f_n(x) = \lim f_{n_{k(x)}}(x)
\end{equation}
2 Maybe $ f_n(x) $ can not converges. But, there are subindices $ n_k $ such that $ f_{n_k}(x)  $ converges.
