Let $s, \delta \in (0,1)$. Consider the outer measure on $\mathbb{R}$, $\mu^s_{\delta}$, defined by
	\begin{align*}
		\mu^s_{\delta}(E):=\inf \left\{\sum_{j}\left|I_{j}\right|^s: E \subset \cup_{j} I_{j}: I_{j} \text { closed intervals }, |I_j|\leq\delta\right\}.
	\end{align*}
For an interval $I \subset \mathbb{R}$, $|I|$ denotes the length of $I$. I want to prove that if $E$ is an interval and $\delta< |E|$, then 
	\begin{align*}
		\mu^s_{\delta}(E) \geq  \delta^{s-1}-\delta^s.
	\end{align*}
I think the key is to express the infimum in terms of $\delta$ and such a cover is a little bit larger than $\delta^{s-1}-\delta^{s}$. I feel like I need to make $\delta$ as large as possible.