(a) State and prove the pumping lemma for regular languages.

(b) Let $D$ be a minimal deterministic finite-state automaton whose language $\mathcal{L}(D)$ is finite. Let $\Gamma_{D}$ be the transition diagram of $D$, and suppose there exists a non-empty closed path $\gamma$ in $\Gamma_{D}$ starting and ending at state $p$.

(i) Show that there is no path in $\Gamma_{D}$ from $p$ to any accept state of $D$.

(ii) Show that there is no path in $\Gamma_{D}$ from $p$ to any other state of $D$.