Problem of OH after class 28

For $S\subset X$ compact and a $x\notin S$
Show $\exists \epsilon >0$, s.t.

$$\underset{y\in S}{\inf }d(x,y)\ge \epsilon >0$$

Then according to $d(x,y)$ is continue, we can say $f(y)=-d(x,y)$ is also continue.
After that we can say

$$(\exists y_0)\underset{y\in S}{\sup }f(y)=f(y_0)$$

is true by Thm 28 OH. According to $x\notin S$, we can say $f(y_0)\ne 0$ is true.
Hence we can take $\epsilon =d(x,y_0)$ and

$$\underset{y\in S}{\inf }d(x,y)\ge \epsilon =d(x,y_0)>0$$

is true. $\square$