Solution
= Solution
The distance function $x\mapsto d(x,c)$ is <continuous> because the triangle inequality gives $|d(x,c)-d(y,c)|\le d(x,y)$. Hence $\overline B(c,r)=\{x:d(x,c)\le r\}$ is the <inverse image> of the closed <interval> $(-\infty,r]$, so is closed. Since it contains $B(c,r)$, it also contains the <closure> of that <open ball>:
$$
\boxed{\operatorname{cl}B(c,r)\subseteq\overline B(c,r).}
$$