Proof of existence of prefix in NFA Let $$A= (Q,Σ,δ,s,F)$$ be a NFA. Prove the following statement:

$∀q ∈ Q: (∃ w_2 ∈ Σ^* :δ (q, w_2) ∩ F \ne  ∅ ⇒ ∃w_1 ∈ Σ^*:w_1w_2 ∈ L(A))$

I figured out that it means that if all the states have a path to the final state, that the word describing their path is a suffix of a word that is accepted from the NEA. 

The thing is: How can I be sure that the NEA accepts all words with that suffix if it doesn't start from the s state.   

Any leads?
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 $w_1$.