If $L = \lim \sup a_n$ and $l = \lim \inf a_n$ then given any $\varepsilon > 0$ there is a natural number $N$ such that $l - \varepsilon < a_n < L + \varepsilon$ whenever $n > N$.
This seems like it would be really easy to prove but I'm having trouble with it. My understanding is that $\limsup a_n = \sup C$ and $\liminf a_n = \inf C$ where $C$ is the set of cluster points of $\{a_n\}$. Since $L$ and $l$ are real numbers $C \neq \emptyset$ and so we can deduce that $\{a_n\}$ is bounded above and below. So it has a supremum and infimum. But I'm just not seeing where to go from here. Any hints or proofs would be greatly appreciated!