I have to prove that a sphere centered at $x\_0$ with radius $r$

$S(x\_0,r) = \{x \in X | d(x,x\_0) = r\}$

is a closed set.

I know I should prove that the complement is open. If I take a point $y$ inside the complement and consider the open ball with radius $\epsilon = \frac{d-r}{2}$ where $d$ is the distance from $x\_0$ to $y$ I have to show that the distance $d(x\_0,y) > r$. This is where I am stuck. I thought about using the triangle inequality, but I am sure how to proceed. Any help would be greatly appreicated.