I am stuck with a question that the constant $3$ in the Vitali covering lemma can not be replaced by any positive constant less than that in the finite case.
Observe that this question is different from the question of why $3$ is considered as bad constant because I am not considering infinite case here.
For example, I was thinking in real line and the argument can be generalised in the higher dimension; if I have $(0,4)$ and constant is $c<3$ then taking the interval $(8,4+\frac{3-c}{2})$ or anything would not give me my answer. So I was doubting the validity of the statement in a metric space but I don't know whether I am missing anything or not!!
You idea is good. Even more simple: a collection consisting of two closed touching balls of equal radius shows that $3$ cannot be replaced by a smaller number.