Prove $\epsilon(S\cap T)\subseteq S \cap T$
Since your definition of $\epsilon$-closure isn't really a definition, it is impossible to prove anything using it. Instead, let me use the following definition: the $\epsilon$-closure of a set $S \subseteq Q$ consists of all states $x \in Q$ which are reachable from a state in $S$ by a (possibly empty) $\epsilon$-path (which is a path consisting of $\epsilon$-transitions).

Suppose that $\epsilon(S) = S$ and $\epsilon(T) = T$, and let $q \in \epsilon(S \cap T)$. Thus $q$ is reachable by a state $r \in S \cap T$ via an $\epsilon$-path. Since $r \in S \cap T$, in particular $r \in S$, and so, by definition, $q \in \epsilon(S) = S$. Similarly, $q \in \epsilon(T) = T$. Therefore $q \in S \cap T$.