This question was raised at a different place without sufficient answers.
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.
The problem is that in the literature, limit point is frequently used but without a definition. So what is the default definition in computer science or optimization? Is Definition 1 the default? For instance, what is the presumed definition of limit point in https://arxiv.org/pdf/1209.2385?