Question about $\liminf$ and $\limsup$ of a sequence Suppose that a sequence $\{x_k\}$ has a clusterpoint (or more..) $c\in\mathbb{R}$. what conclusion, if any, can be drawn about either $\liminf x_k$ or $\limsup x_k$ ? 
I don't know the conclusion... I think that when the sequence has only one cluster point 
then $\liminf x_k= \limsup x_k = c$. But if the sequence has many cluster points, I can't draw conclusion about  either $\liminf x_k$ or $\limsup x_k$.
 A: There are several different (but equivalent) ways to define $\limsup$ and $\liminf$, so it all depends on which one you are familiar with.
One definition is that $\limsup x_k$ is the supremum of all points that are limits of subsequences of $x_k$. That is,
$$\limsup x_k = \mathrm{sup}\Bigl\{ a\in\mathbb{R}\cup\{\pm\infty\}\Bigm| \text{there is a subsequence of }x_k\text{ that converges to }a\Bigr\}$$
(where we think of a subsequence as "converging to $\infty$" or "to $-\infty$" if the limits are equal to $\infty$ or $-\infty$). 
Similarly, one can define $\liminf x_k$ to be the infimum of all points that are limits of subsequences of $x_k$.
Note that a real number $a$ is a limit of a subsequence of $x_k$ if and only if $a$ is a cluster point for the sequence $x_k$. Given this, the fact that $c$ is a cluster point of $x_k$ tells you that $c$ belongs to that set for which $\limsup x_k$ is the supremum, and $\liminf x_k$ is the infimum. So what conclusion can you draw then?
If you have other definitions of $\limsup$ and $\liminf$, then please state the ones you know explicitly in your question.
A: In general you have $\liminf x_k \leq c \leq \limsup x_k$. Note that $\liminf x_k$ may be strictly smaller than any of its clusterpoints ( $x_k = -k$ which even has no clusterpoints at all ) and a similar result obviously holds for $\limsup x_k$.
