Given a positive real number $\alpha$ and a positive rational number $p/q$ in reduced form let's define the quality of $p/q$ as an approximation to $\alpha$ as$$-\log_q|\alpha - p/q|$$ I'm looking at the sequence of approximations to the golden ratio $(F_{n+1}/F_n)_{n=3}^\infty$ where $F_n$ is the $n$'th Fibonacci number. The quality monotonically decreases towards 2. If we subtract 2 and take the reciprocal we approach an arithmetic sequence with $\Delta = 0.597987\dots$
... at least that's what seems to be happening looking at the first 10 or so terms. My question is how do you prove this limit exists? More explicitly, I mean $\lim_{n\to\infty}s_{n+1}-s_n$ where $$s_n = \frac1{-\log_{F_{n}}(|\Phi-F_{n+1}/F_n|)-2} = -\frac{1}{\log_{F_n}(|F_n^2\Phi-F_nF_{n+1}|)}$$
Using the recurrence relationship one can show $F_n\varphi -F_{n+1}=(-1)^n \varphi^{-n}$, so that $$ s_{n+1}-s_n = \frac{\log F_n}{\log (F_n\varphi^{-n})}-\frac{\log F_{n+1}}{\log (F_{n+1}\varphi^{-n-1})} $$ Then using a well known asymptotic as $n \rightarrow \infty, \space F_n\varphi^{-n}\rightarrow \frac{1}{\sqrt{5}}$ so that $$s_{n+1}-s_n \rightarrow \frac{1}{(-\log {\sqrt{5}})} \log {\frac{1}{\varphi}} \approx 0.597987435665$$