Hausdorff dimension of a countable set

Question

I don't understand why the Hausdorff dimension of a countable set in $\mathbb{R}^n$ is $0$. Can someone please give me a hint?

Thank you!

Answer 1

Hausdorff dimension is defined as the inf of those $d$ such that the $d$-dimensional Hausdorff measure vanishes. Measures are countably additive, so any measure which vanishes on all singletons vanishes on all countable sets; and it is fairly obvious that Hausdorff measures of all dimensions $d>0$ vanish on a singleton. So, for a countable set just as for a singleton, the inf is $0$.

If you want a more concrete way of looking at it, if $C = \{x_n : n\in\mathbb{N}\}$ is countable, then $C$ can be covered by a sequence of balls centered on the $x_n$, whose radii are any sequence of positive real numbers, and in particular, one which tends arbitrarily rapidly to $0$ (so in particular, one such that the $d$-powers of the diameters of the balls converges to an arbitrarily small real number).