# Use the pumping lemma to show it's not regular

I just learned pumping lemma this week and got confused on this question.

B={$$a^{fn}$$ | $$f_n$$ is a Fibonacci number} for $$a \in Σ$$.

Hint: the sequence of Fibonacci numbers get increasingly further apart.

I am trying to prove this is not regular using pumping lemma, but 'Fibonacci numbers get increasingly further apart' seems really tricky when I try to pump up to prove it's not $$\notin$$ B.

Suppose that $$B$$ were regular. According to the pumping lemma, there exists an integer $$p \geq 1$$ such that every word $$w \in B$$ of length at least $$p$$ has a decomposition $$w = xyz$$, with $$|xy| \leq p$$ and $$y \neq \epsilon$$, such that $$xy^tz \in B$$ for all $$t \geq 0$$.
Pick some Fibonacci number $$F_n$$ such that (1) $$F_n \geq p$$ and (2) $$F_{n+1} - F_n > p$$. Then $$1^{F_n} \in B$$ has length at least $$p$$, and so it can be decomposed as $$1^{F_n} = xyz$$, where $$|xy| \leq p$$ and $$y \neq \epsilon$$, and $$xy^2z \in B$$. But $$xy^2z = 1^m$$, where $$F_n < m \leq F_n + p < F_{n+1}$$.