Limit superior of a sequence is equal to the supremum of limit points of the sequence?

The problem is that you’ve misunderstood the notion of limit point of a sequence. I prefer the term cluster point in this context, but by either name it’s a point $x$ such for any nbhd $U$ of $x$ and any $n\in\mathbb{N}$ there is an $m\ge n$ such that $x_n \in U$. Thus $1$ and $-1$ are both cluster points of the sequence $\langle (-1)^n:n\in\mathbb{N}\rangle$, though this sequence doesn’t converge to anything.

Equivalently $x$ is a cluster point of $\langle x_n:n\in\mathbb{N}\rangle$ iff for each nbhd $U$ of $x$, $\{n\in\mathbb{N}:x_n \in U\}$ is infinite. If the space is first countable, $x$ is a cluster point of $\langle x_n:n\in\mathbb{N}\rangle$ iff some subsequence of $\langle x_n:n\in\mathbb{N}\rangle$ converges to $x$.