Sequence $ \{ f_n(a) \}$ satisfies $ \displaystyle f_{n+1}(a) = 2 - \frac{a}{f_n(a)}$, $ f_1(a) = 2$, $ n=1,2, \cdots$. If there exists a natural number $ n$, such that $ f_{n+k}(a) = f_{k}(a), k=1,2, \cdots$, then we call the non-zero real $ a$ a $ \textbf{periodic point}$ of $ f_n(a)$. Prove that the sufficient and necessary condition for $ a$ being a $ \textbf{periodic point}$ of $ f_n(a)$ is $ p_n(a-1)=0$, where $ \displaystyle p_n(x)=\sum_{k=0}^{\left[ \frac{n-1}{2} \right]} (-1)^k C_n^{2k+1}x^k$, here we define $ \displaystyle \frac{a}{0}= \infty$ and $ \displaystyle \frac{a}{\infty} = 0$.