I'm supposed to proof that the sequence $x_n:=f_{n+1}/f_n,n\in\mathbb{N}$ converges to the golden ratio $\Phi$ where $f_n:=f_{n-1}+f_{n-2}$ is the Fibonacci sequence with $f_1=1$ and $f_2=1$. At first I needed to show that
\begin{align*} |\Phi-x_n|&=\left|1+\frac{1}{\Phi}-\left(1+\frac{1}{x_{n-1}}\right)\right|\\ &=\left|\frac{1}{\Phi}-\frac{1}{x_{n-1}}\right|\\ &=\left|\frac{x_{n-1}-\Phi}{\Phi x_{n-1}}\right| \end{align*}
but I'm stuck trying to show that $|\Phi-x_{n}|\leqslant|\Phi-x_2|/\Phi^{n-2}$. Can anyone please help me to solve this problem? Thanks in andvance!