In vector analysis the closedness of a set can be defined as
A set A is closed, iff for each convergent sequence $x_k \in A$, the limit point $a$ of the sequence $x_k$ also belongs to A.
What does it mean for a set to contain "all convergent sequences $x_k \in A$"? Why is there the wording "for each convergent sequence $x_k \in A$"?
Can one actually list "all convergent sequences $x_k \in A$"?
It means that you consider all sequences $x_n$ that are convergent [in the underlying vector space] so that all of its elements $x_n$ lie in $A$.
For example, the set $A = (0, 1)$ is not closed in $\mathbb{R}$, because the sequence $x_n = \frac{1}{n}$ is convergent [in $\mathbb{R}$], but its limit $0$ isn't contained in $A$.