TITLE: If $a\in \mathrm{clo}(S)$, does it follow that there exists a sequence of points in $S$ that converges to $a$?
QUESTION [6 upvotes]: Let $X$ be a topological space and $S=\{x_n\}$ be a sequence of points in $X$. Suppose $a$ is a point in $X$ such that $a$ is adherent to $S$(that is $a$ is in the closure of $S$),I want to ask if there must exist a sequence $\{y_n\}$ in $S$ such that the limit of $\{y_n\}$ is $a$. If not,please give an example,Thanks!

REPLY [2 votes]: For another example, let $X$ be the Stone–Čech compactification of $\Bbb{N}$. It does not contain any subspace homeomorphic to $A(\aleph_0)$, i.e., in $\beta \mathbb{N}$ there are no non-trivial convergent sequences (see Engleking's book Corollary 3.6.15).