I'm trying to show that when $\mathcal{H}^a(K) > 0$ and $b < a$ then $\mathcal{H}^b(K) = \infty$. I did this by considering a cover $K \subset \{U_i\}$ of radius $\delta$. Then $$\sum|U_i|^a = \sum|U_i|^{a-b}|U_i|^b \leq \delta^{a-b}\sum|U_i|^b$$This implies $\left( \frac 1 \delta\right)^{a-b}\mathcal{H}_\delta^a \leq \mathcal{H}_\delta^b$. Taking $\delta \rightarrow 0$ gives the result.
I'm uncertain about my work because of the way $\delta$ is being used to take limits in the inequality. Is this okay?