Vtyjkyuk

Let $E subseteq mathbbR^d$. Let $epsilon > 0$. Show that there
exist some countable covering $Q_k$ such that

$$|E|_e leq sum_k textvol(Q_k) leq |E|_e + epsilon$$

I need a hint on how to proceed with such a proof, not necessarily a full solution. I think there is some property of infimums that lets me "squeeze" in numbers, but I don't know what that property is.

If $|E|_e = infty$, pick a trivial covering that diverges. Otherwise, since

$$
|E|_e = inf left sum_n geq 1operatornamevol(Q_k) : Q_k_k geq 1 text covers $E$ right
$$

for any $varepsilon > 0$, there exists some covering $Q_k_k geq 1$ so that

$$
0 leq sum_n geq 1operatornamevol(Q_k) - |E|_e < varepsilon
$$

which is exactly what you have to prove.

The property we are using is the following, assuming that you've defined the infimum as the 'larger' lower bound of a set:

Proposition. Let $A subseteq mathbbR$ and $iota := inf A$. Then, for any $varepsilon > 0$ there exists $a in A$ so that $0 leq a - iota < varepsilon$.

Proof. Let $varepsilon > 0$. Since $iota < iota + varepsilon$, the latter cannot be a lower bound for $A$ because $iota$ is an infimum. Thus, there exists $a in A$ so that $a < iota + varepsilon$. Therefore,

$$
0 leq a - iota < varepsilon
$$

as desired, with the first inequality given by the fact that $iota$ is by construction less or equal than $a$. $square$