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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?

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    $\begingroup$ One definition is for a limit point of a sequence; the other for a limit point of a set. $\endgroup$
    – saulspatz
    Apr 25, 2018 at 16:31

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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.

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