"Let $A$ be a subset of a complete metric space. Assume that for all $ε > 0$, there exists a compact set $A_ε$ so that $∀ x ∈ A, d(x, A_ε)<ε$. Show that $A$'s closure is compact."
I am trying to prove it with "Cauchy sequences in complete metric space are convergent" and "sequential compact set is compact". But I do not know how to prove all sequence in A(or A's closure?) has a cauchy subsequence.
Any help or advice would be greatly appreciated. Thank you in advance!