closed and discrete subset of a metric space

Question

Is every closed and discrete subset of a metric space uniformly discrete?

I tried searching for a counterexample but could not find any.

Answer 1

Let $X$ be the metric space $(0,1]$ in the usual metric.

Then $A = \{\frac1n: n =1,2,3,\ldots\}$ is closed and discrete but not uniformly so.