Showing that some point is in a compact set

Question

$(X, d)$ is a metric space. $W$ is a compact subset of $X$.

Some $x \in X$ satisfies that $$ \forall \epsilon > 0: \exists y \in W: d(y, x) < \epsilon $$

Then, can we conclude that $x \in W$?

If the proposition is modified to $\exists y \in W:y\neq x \land d(y,x) < \epsilon$, the point $x$ is a limit point of $W$, hence in $W$.

Answer 1

For each $n\in\Bbb N$, there is some $w_n\in W$ such that $d(w_n,x)<\frac1n$. Therefore, $x=\lim_{n\to\infty}w_n$, and so $x\in\overline W=W$, since $W$ is compact and therefore closed.