Definition 1: We say that a vector $x \in R^n$ is a limit point of a sequence $\{x_k\}$ in $R^n$ if there exists a subsequence of $\{x_k\}$ that converges to $x$. This definition appears frequently in the optimization literature, for instance, see Bertsekas, Nonlinear Programming, 2nd edition, page 666.
But a definition of limit point in real analysis is different.
Definition 2: A point $z_0$ is a limit point for a set of point if every neighborhood of $z_0$ contains points, other than $z_0$ of set.
Accordingly, $a_n=\{5, 4, 3, 2, 1, 0, 0, ...,\}$ has a limit point of $0$ based on the first definition, but $0$ is not a limit point based on the second definition.
Is it just me confused?
It is common to find different "definitions" of the same term in different texts. You can find real analysis texts which define natural numbers as $\{0,1,2,\ldots \}$, and just as well you can find real analysis texts which define natural numbers as $\{1,2,3,\ldots\}$.
The important thing is that within a given text, you should take a definition as gospel truth, as all further development should be based on that definition. Maybe a given text says that "$X$ means $Y$", while in your experience you learned "$X$ means $Z$". When you are lucky, $Y$ and $Z$ are in fact equivalent properties that are stated in different terms. However sometimes there might be significant differences between $Y$ and $Z$.
That said, your example does not make much sense. Writing $a_n = \{5,4,3,2,1,0,\ldots,0\}$ implies that $a_n$ has some final element. Sequences are generally presumed to be indexed by some infinite set, and have no final element.