I'm going to prove that the limit $\lim_{n\to\infty}{\frac{F_{n+1}}{F_n}}$ exists.
Proof: We will prove that the sequence is a cauchy sequence. We have that: $$ \left|\frac{f_{n+1}}{f_n}-\frac{f_n}{f_{n-1}}\right|= \left|\frac{f_{n+1}f_{n-1}-(f_n)^2}{f_n f_{n-1}}\right|= \left|\frac{(-1)^n}{f_n f_{n-1}}\right|\leq \frac{1}{f_n f_{n-1}}. $$
We have that $\frac{1}{f_n f_{n-1}}$ is a subsequence of $\frac{1}{n}$ and converges to 0. Therefore our sequence is a cauchy sequence and we are done.
My question is if my argument that this is a subsequence of $\frac{1}{n}$ works and proves that the fibonacci ratio sequence is a cauchy sequence.
