Probably the easiest way to think about this is to consider an equivalent statement for the real numbers. Consider the range $[0,1]$, for instance. You're given a finite set of real numbers within that range. If you think about these values on the number line, it should be fairly obvious that there are necessarily points that are a finite distance away, so I cannot use members of this set as arbitrarily accurate approximations for any real number in the range. (In this case, if your set contains $n$ elements, it would be best to have them at values $(2i-1)/(2n)$ for $i=1$ to $n$ so that all real numbers in the range are within $1/(2n)$ of some element in the set. But if you try to bunch some closer together, obviously other have to get further apart).
You can make the same argument for any continuous space for which there is a distance measure.