Every finite language is regular

Define a machine $D$ with state set $P\cup \{\mathrm{\perp }\}$, start state $\epsilon$, accepting set $F=L$, and transition rule $\delta (p,a)=\{\begin{array}{ll}pa,& \text{if }pa\in P,\\ \mathrm{\perp },& \text{if }pa\mathrm{\notin }P,\end{array}\delta (\mathrm{\perp },a)=\mathrm{\perp }\mathrm{.}$ By [F1] the state set is finite, and the transition rule is total on $(P\cup \{\mathrm{\perp }\})\times \mathrm{\Sigma }$.

We claim that, for every word $w$, the run from $\epsilon$ ends at the state $w$ when $w\in P$, and ends at $\mathrm{\perp }$ when $w\mathrm{\notin }P$. This follows by induction on the length of $w$: appending a letter keeps the machine at the longer prefix exactly when that longer word still lies in $P$, and otherwise sends the run to $\mathrm{\perp }$, where it remains.

A word $w$ is accepted by $D$ exactly when the final state lies in $F=L$. By step 1.2 this happens exactly when $w\in L$, because every word in $L$ is one of its own prefixes and therefore belongs to $P$.

Thus $L=L(D)$ for the DFA from step 1.1, so [L1] shows that $L$ is regular.