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.
 A: Your definitions are good, but I usually prefer this definition of $\limsup$:
$$\limsup_{n\to\infty} y_n=\lim_{n\to\infty}\sup_{k\ge n} y_k$$
the point being that $\sup_{k\ge n} y_k$ decreases with $n$ (it is a supremum over smaller and smaller sets), so it has a limit, or converges to $-\infty$.
It is a good exercise to show that there always exists a subsequence converging to $\limsup y_n$, and that the limit of any convergent subsequence is at most as large as $\limsup y_n$.
For functions, the $\limsup$ is defined pointwise, just as for limits.
Oh, and why are subsequences needed? A trivial example is $y_n=(-1)^n$. It has no limit, but the superior limit is $1$, and the inferior limit is $-1$.
A: Question $(2)$:
I think one thing that gets confusing, when referring to subsequences of a sequence, is the use of indices to denote them, e.g. when using a "double subscript" to denote a subsequence $f_{n_k}(x)$ of $f_n(x)$, or a subsequence $\{a_{n_k}\}$ of $\{a_n\}$.
Perhaps an example of a subsequence would help illustrate what is meant by a subsequence:
Let, e.g., $a_n = 1, \frac{1}{2}, \frac{1}{3}\dots \frac{1}{n}$.
We can define a subsequence $\{a_{n_k}\} \text{ of}\; \{a_n\}$ where $a_{n_k}$ is defined by $a_{2n} = \frac{1}{2}, \frac{1}{4}, \dots \frac{1}{2n}$.

The important thing to note is that a sequence $\{b_n\}$ is a subsequence of $\{a_n\}$ if and only if for each $j\ge 1 $ both of the following hold:
$(1)$ $b_j$ is one of the terms of the sequence $a_1, a_2, \dots$ and 
$(2)$ the term $b_{j+1}$ appears later than $b_j$ in the sequence $a_1, a_2, \dots$

Note:
It may happen that sequence $\{a_n\}$ does not converge, while some subsequence(s) of $\{a_n\}$ does converge. But a sequence $\{a_n\}$ converges if and only if every subsequence of $\{a_n\}$ converges.
A: 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.
