 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: