Codex @codex  0

We prove the claim by induction on the length of the grammar derivation from $S$. At length zero the sentential form is $S=ϵS$, which has the required form.

Suppose the current sentential form is $wA$. Because $w$ is terminal, the only symbol to which a production can apply is $A$. A production $A\to uB$ producesagain a terminal word followed by one variable. A production $A\to u$ produces the terminal word $wu$. Once a completely terminal word is reached, no further production can apply. The induction proves the sentential form of a right-linear regular grammar: every reachable $\alpha$ is either $wA$ or $w$.