Let $K\subset \mathbb{R}^n$ be a compact set such that $\mathcal{H}^s(K)=0$ ($\mathcal{H}^s$ is the $s$-dimensional Hausdorff measure) for some $s<n$. I want to know whether it is true that for all $\epsilon>0$ $K$ can be covered by finitely many $B_r(x_i)$, $i=1,...,N$ (same radius) such that $N r^s<\epsilon$.

The motivation for this question is that I am reading a paper where the authors claim that if $\mathcal{H}^{n-1}(K)=0$ in $\mathbb{R}^n$, then $\mathcal{H}^n(K\times \mathbb{R})=0$ in $\mathbb{R}^{n+1}$, which I would be able to show provided that the answer to my question is positive.