Solution
= Solution
For a fixed centre $a$, the triangle inequality gives $|d(a,x)-d(a,y)|\leq d(x,y)$, so the distance-to-centre <function> is <continuous>. Its <preimage> of $(-\infty,r]$ is the closed ball, hence is closed. Since that <closed set> contains the corresponding open ball, it contains its closure:
$$
\boxed{\overline{B(a,r)}\subseteq\overline B(a,r)}.
$$
Here the bar on the right denotes a closed ball, while the bar on the left denotes closure.