Show that $\limsup \{a_n\}$ is the largest cluster point.
Definition: We call an extended real number a cluster point of a sequence $\{a_n\}$ if a subsequence converges to this extended real number.
I'm not sure how to 'unwrap' the definition of a cluster point to prove this.
This was my attempt. Let $l = \limsup \{a_n\}$. Suppose that there is an $x$ such that $x > l$ and it is a cluster point of $l$. If we let $\epsilon>0$, then $x-\epsilon > l$. There exists $n \geq N$ such that $x_n > x-\epsilon$. Let $\tilde{n}=\min\{n \geq N \text{ such that } x_n > x-\epsilon \}$.
I'm not really sure where to go from here.