Limsup/inf of sequences of functions Does anyone have a good, intuitive explanation for what the $\limsup$ and $\liminf$ of a sequence of functions is? So, if I have a sequence of functions $f_n(x)$, what is the $\limsup$ of that sequence?
I am particularly looking at the sequence $f_n = n \cdot \chi_{[1/n,2/n]}$, but am really interested in seeing a more general explanation.
 A: Perhaps a graphical visualization of these two concepts would be helpful:

As for your function sequence, it looks something like

Clearly both limits are zero: $\limsup \,f_n = 0$, $\liminf \,f_n = 0$.
A: Just remember that for a sequene $a_n$, the limsup is the largest limit of a convergent subsequence, and the liminf is the smallest limit of a convergent subsequence. For functions $f_n(x)$, you just look at it point by point.
So for $f_n = n \chi_{[1/n,2/n]}$ it's actually much easier because the limit exists. For a fixed $x$, for all $n$ large enough, $x \notin [1/n,2/n]$, thus $\lim f_n(x) = 0$ for all $x$.
An example of where the limsup/liminf exist but the limit do not would be something like the sequence
$$
1_{[0,1)},
$$
$$
1_{[0,1/2)}, 1_{[1/2,1)},
$$
$$
1_{[0,1/3)}, 1_{[1/3,2/3)}, 1_{[2/3,1)},
$$
$$
\cdots
$$
$$
1_{[0,1/k)}, \ldots, 1_{[i/k,(i+1)/k)}, \ldots, 1_{[(k-1)/k, 1)},
$$
$$
\cdots
$$
This is the standard "dancing" example.
Then $\limsup_n f_n(x) = 1$ for all $x$ and $\liminf f_n(x) = 0$ for all $x$, and hence $\lim_n f(x)$ does not exist for any $x \in [0,1]$.
