Proof of existence of prefix in NFA
The statement is false. Consider the NFA on the alphabet $\{a\}$ having two states: an initial state $q_0$, and the unique final state $q_f$. The transition function is given by $\delta(q_0,a) = \{q_0\}$ and $\delta(q_f,a) = \{q_f\}$. For every word $w_2$ it is the case that $\delta(q_f,w_2) \cap F = \emptyset$, yet $w_1 w_2 \notin L(A)$ for all $w_1$.