Why is 3 a bad constant in the Vitali covering lemma (infinite version)?

Question

The Hardy-Littlewood maximal function inequality can be proved using the Vitali covering lemma (infinte version) with constant k=5. Actually it can be shown that the constant in the lemma just needs to be k>3. But I cannot find a counter example why 3 is not enough for the infinite case (it is enough in the finite version of the lemma). Has anyone seen such a counter example?

Answer 1

$$ Z=\{ B (x,r) : x, r \in R,|x| < \frac {1}{2}, |x| < r <(|x| + 1)\frac {1}{3} \}$$

Then every ball in $Z$ contains $0$ and hence every disjoint subcollection of $ Z$ consists of just one ball. But for any ball $B(x, r)$ , the expanded ball $3B(x, r)$ does not cover $Z = (-1,1) $