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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. $\epsilon(S) \supseteq S$....

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