# Characterization of lim sup, lim inf

If $(a_n)$ is a real sequence, in lecture we had:
\begin{align}\limsup_{n\to\infty} a_n=a \iff &(i)\forall \epsilon >0 \,\exists n_0\in \mathbb{N} :a_n<a+\epsilon \forall n\ge n_0\\\text{ and }&(ii) \forall \epsilon >0 \, \forall m\in \mathbb{N} \, \exists n\ge m :a_n>a-\epsilon\end{align} and such a analogue characterization for lim inf.

I know the definition of $\limsup\limits_{n\to\infty} a_n=\sup H(a_n)$ with $H(a_n)$ set of all the limitpoints of $(a_n)$. I don't understand the epsilon-characterization of lim sup an, maybe you can draw a picture, explain it in words why it is equivalent (or prove it directly, but I only want to understand it). Later I want to do an example if I have understand this. Maybe you want to help? Regards

• There is also a third commonly used definition: $\limsup a_n= \lim_{n\to\infty} \sup_{k\ge n} a_k = \inf_{n\in\mathbb N} \sup_{k\ge n} a_k$. May 2, 2014 at 22:03
• thank you. I will try to understand it May 2, 2014 at 22:55
• This is (to some extent) related: math.stackexchange.com/questions/281807/… May 2, 2014 at 23:06
• @RedRose could you please give the corresponding characterizations for the liminf? Thank you in advance! Jan 16, 2021 at 22:15