Given a succession $r(n)= 1 +\frac{1}{r(n-1)}$
where $r (1)=1$ and golden number $\phi =\frac{1+\sqrt{5}}{2}$.
How do I prove that $$\left\lvert r(n)-\phi\right\rvert \leq \frac{1}{\phi ^n}\quad\mbox{and}\quad r(n)\xrightarrow[n\to\infty]{}\phi? $$
This knowing that absolute value of (( r(n)-golden number)/(r(n-1) - golden number)) is less or equal to (1/(golden number)).
Hint:
Once you've proven that
$$\frac{|r(n)-\phi|}{|r(n-1)-\phi|} \leq \frac{1}{\phi},$$
prove that $|r(1)-\phi|\leq \frac{1}{\phi}$ and use induction. For the second part, use an $\epsilon$-$\delta$ argument once you know part 1.