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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. $\epsilon(S) \supseteq S$....
So it has to be something like this $w=b{a}^{2m+2}b^mc^m$. $b{a}^{2m+2}b^mc^m$ is not the general form for $w=bxc^m\in S$. If $x$ must be in the form of $a^*b^*$, $w$ will be something like $b{a}^{m+n+k+2}b^nc^m$ for some $n,k\ge0$. Just for completeness, $w$ could be like $baba^{m+1}c^m$ or $baba^{m+1}a^kb^kc^m$ or many other forms. Of course, it is ...