True or false. In an infinte metric space, an open ball of radius 2 is an infinte set.

Well, my answer is false,

Let M= R.

Let the metric be given by :

d(x,y) = 0 if x=y

d(x,y) = 2 if x<>y

Then in this metric space, the open ball B[a;r] about a in M has just a single point.

Is my reasoning right??