Your definition of $\epsilon$-closure is quite problematic. Here is a better formulation:
$\epsilon(S)$ is the intersection of all sets $T \subseteq Q$ such that (i) $T \supseteq S$ and (ii) if $q \in T$ then $\delta(q,\epsilon) \subseteq T$.
Here is a series of claims which imply $\epsilon(S) = \epsilon(\epsilon(S))$.
Claim 1. $\epsilon(S) \supseteq S$....