How can I prove a closed ball is closed on metric topology?

Question

Let $(X,\tau _d)$ is metric topology. Show that $B_r[x]$ is closed

$X\setminus B_r[x]$ must be open set. It is open iff $X\setminus B_r[x] \in \tau _d$ iff $\forall y \in X\setminus B_r[x] , \exists \epsilon >0 , B_\epsilon (y) \subseteq X\setminus B_r[x] $

Assume $ z \in B_\epsilon (y) $ $d(y,z) \lt \epsilon$ and we know $d(x,y) \gt r$ from $y \in X\setminus B_r[x]$

How can I show $d(x,z) \gt r$ from inequalities??

Answer 1

Let K be the closed ball about a of radius r.
f(x) = d(a,x) is continuous. I = [0,r] is closed.
Thus $K = f^{-1}(I)$ is closed.