Limit of a recursion sequence

[elim]

Let $ y_{0}\geq 2,\; y_{n}=y_{n-1}^{2}-2, \quad \displaystyle{s_{n}=\frac{1}{y_{0}}+\frac{1}{y_{0}y_{1}}+\cdots + \frac{1}{y_{0}y_{1}\cdots y_{n}} }$
prove that : $\displaystyle {\lim_{n\to\infty} s_n=\frac{1}{2}\left(y_0-\sqrt{y_0^2-4}\right)} $