This comes from a book on geometric measure theory in a chapter introducing the Hausdorff measure $\mathcal{H^t}$. I cannot see in this proof how $\sum_i d(E_i)^s \leq \mathcal{H^s_{\delta}}(A)+1$ is obtained.
is my definition of the Hausdorff measure.
By definition $\mathcal H^s_\delta(A)=\inf\{\sum_id(E_i)^s: A\subset \cup_i E_i, \> d(E_i)<\delta\}$
Therefore, by definition of infimum, for any positive number $K$, there must exist a family $E_i$ with $d(E_i)<\delta$, $A\subset U_i(E_i)$ and $\sum_i d(E_i)^s<\mathcal H_d^s(A)+K$. Set $K=1$.