Proving the equivalence of two definitions of NFA acceptance
Just a hint: 


First, generalize the problem: show that, for any $q\in Q$ and any word $w$, $q \in\hat\delta(S,w)$ if and only if $\exists r_0,...,r_{|w|}\in Q$ such that $r_0\in S$, $r_{|w|}=q$, and $r_{i+1}\in\delta(r_i, w_{i+1})$.
Second, show this statement by induction on the length $|w|$ of $w$. This is trivial for the base case, and should be easy for the inductive case, just apply the induction hypothesis and the definition of $\hat\delta$.
Finally, show that your equivalence follows from the general statement: apply it for any $q\in F$.