Let $H$ be a complete metric space, we equip $H$ with the induced metric norm. Let $S$ be a set of $H$.
Assume that for any $\epsilon>0$, we can find compact set $K_\epsilon$, such that $S$ is contained in a $\epsilon$ neighborhood of $K_\epsilon$.
Q Can we say $\bar S$, i.e. the closure of $S$, is compact?