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

I'm taking your definition of limsup to be the "largest subsequential limit".  
i) says that no subsequential limit can be bigger than the limsup.  
ii) says that a subsequential limit can be at least as big as the limsup.  
