9. $[7]$ Let $a_{0}=\frac{6}{7}$, and
$$
a_{n+1}=\left\{\begin{array}{ll}
2 a_{n} & \text { if } a_{n}<\frac{1}{2} \\
2 a_{n}-1 & \text { if } a_{n} \geq \frac{1}{2}
\end{array}\right.
$$

Find $a_{2008}$.
$$
a_{1}=\frac{5}{7}, a_{2}=\frac{3}{7}, a_{3}=\frac{6}{7}=a_{0}
$$

So this sequence repeats every three terms, so $a_{2007}=a_{0}=\frac{6}{7}$. Then $a_{2008}=\frac{5}{7}$.