I would like to prove that any context free language over a 1 letter alphabet is regular. I understand there is Parikh's theorem but I want to prove this using the work I have done so far:
Let $L$ be a context free language. So, $L$ satisfies the pumping lemma. Let $p$ be $L$'s pumping constant. Let $L = L_1 \cup L_2$ where $L_1$ consists of $w$ where $|w| < p$ and and $L_2$ consists of $w$ where $|w| \ge p$. We have a single letter alphabet and since $L_1$ has a restriction on the length of its words, $L_1$ is a finite language. Finite languages are regular so $L_1$ is regular. If I can show that $L_2$ is regular, I can use the fact that the union of regular languages is regular. But I am struggling on showing that $L_2$ is regular. I know that since $w \in L_2$ has to satisfy $|w| \ge p$, by the pumping lemma, $w$ can be written as $w = uvxyz$ where $|vxy| \le p$, $|vy| > 0$ and $\forall t \ge 0$, $uv^txy^tz \in L$. Since we have a single letter alphabet (say the letter is $a$), $uv^txy^tz = uxz(vy)^t = uxz(a^{|vy|})^t \in L$. Now what?