I'm trying to show that limsup $\displaystyle a_n=inf${$\displaystyle sup S_n|n \in N$} where $\displaystyle S_n=${$\displaystyle a_k|k \geq n$} and $\displaystyle a_n$ is a bounded sequence using only the fact that limsup $\displaystyle a_n=sup(T)$, where T is the set of all subsequential limits of $\displaystyle a_n$.

I realize that what I'm asking is a "definition," but I'm just trying to find a way to derive it, basically. Any help on this?