There exists a countable collection of sets $E_i$ with open union such that $\text{diam }E_i<\epsilon$ and $A \subset \bigcup E_i$

Question

On the way to proving that Hausdorff Measure is $G_\delta$ regular (that is for any set $A\subset \mathbb{R}^n$ there is a $G_\delta$ set B that $H^s(B)=H^s(A)$). I found myself stuck on the following problem:

Given an $\epsilon>0$, I need to produce (through construction or proving that it exists) a collection $\{E_i\}_{i\in\mathbb{N}}$ such that

• $\text{diam }E_i < \epsilon$ for all $i \in \mathbb{N}$
• $A\subset \bigcup_{i\in \mathbb{N}}E_i$
• $\bigcup_{i\in \mathbb{N}}E_i$ is open.

EDIT: we must also ask (and this is the tricky bit) that:

• $H_\epsilon^s(A) \leq \sum(\text{diam }E_i)^s\leq H_\epsilon^s(A)+\epsilon$

I cannot think of a way to prove the existence of such a collection. Any help is appreciated.