Cover of a metric space.

Question

Let $E$ be a separable and complete metric space. Let $\epsilon > 0.$ I want to find a cover of $E$ consisting of balls $B(q_n, \epsilon), n \in \mathbb{N}, q_n \in E,$ such that every $e \in E$ is contained only in finitely many balls $B(q_n,\epsilon).$ Is this possible?

Answer 1

I suppose that there is a more elementary proof than this one, but here it goes: every metric space is paracompact (this was proved by A. H. Stone) and every paracompact space is metacompact.