Question 1.11: Prove that every NFA can be converted to an equivalent one t...

Let  $N = (Q,\Sigma ,\delta ,q_{0}, F )$ be any NFA. Construct an NFA $N^{\prime }$ with a single accept state that recognizes the same language as N. Informally, $N^{\prime }$ is exactly like N except it has ε-transitions from the states corresponding to the accept states of N, to a new accept state, $q_{accept}$. State $q_{accept}$ has no emerging transitions. More formally, $N^{\prime } = (Q\cup \left\{q_{accept} \right\}, \Sigma ,\delta ^{\prime }, q_{0},\left\{q_{accept} \right\} )$, where for each q ∈ Q and a ∈ $\Sigma _{\epsilon }$

and $\delta ^{\prime } (q_{accept} ,a)= \phi$ for each  $a\in \Sigma _{\epsilon }$