When does the Hausdorff measure equal infinity?

Question

I'm writing a project on fractals and I got as far with Hausdorff dimension to say that, for some surface $S \subset \mathbb{R}^n$ and $r,s\in \mathbb{R}_{\ge 0}$ such that $r>s$ that $$0 \leq H^r(S) \leq \lim_{\delta \to 0^+} \delta^{r-s}H_\delta^s(S).$$ Naturally, this would imply that $H^r(S) = 0$ whenever $H_\delta^s(S) < \infty$. My layman's question is, how can we show that $H_\delta^s(S) = \infty$ for some $s$? I know that a disk in $\mathbb{R}^2$ would have $H_\delta^s(S) = \infty$ for $s \in (0, 2)$, for instance, but how can I prove that only knowing that $H_\delta^s(S) = \inf \{ \sum_{i} \text{diam}(U_i)^s \}$?