Finding a bound on a sequence defined as ratios of the Fibonacci sequence

Question

I have a sequence $(a_n)$ defined such that $a_n = F_n/F_{n+1}$, where $F_n$ is the $n$th Fibonacci number. I want to show that there exists a real $b \in (0,1)$ such that: $$|a_{n+2} - a_{n+1}| \leq b |a_{n+1} - a_n|$$ holds for all $n \geq 1$. I've been able to rearrange this inequality to: $$\frac{1}{|(1+a_n)(1+a_{n+1})|}\leq b$$ but here I get stuck. I recognize that the terms in $(a_n)$ are not uniformly increasing or decreasing, i.e. the subsequence $(a_{2n})$ is increasing and the subsequence $(a_{2n+1})$ is decreasing, but I don't know if this is useful. Any help?