Example 2 in Chapter 5 of  constructs, for an arbitrary closed subset of the real line, a sequence whose set of limit points is exactly the original closed set. We use a similar construction to show that an arbitrary nonempty closed set in a separable metric space is always the set of limit points of some sequence. We note further that if all nonempty closed subsets of a metric space can be realized as sets of limit points of sequences, then that metric space is separable.
"Limit Sets and Closed Sets in Separable Metric Spaces." Missouri J. Math. Sci. 21 (2) 78 - 82, May 2009. https://doi.org/10.35834/mjms/1316027240