TITLE: Prove that the limit of two consecutive fibonacci numbers EXISTS.
QUESTION [7 upvotes]: Using the definition of Fibonacci numbers, $F_n=F_{n-1}+F_{n-2}$, I can prove that the limit of $\frac{F_{n+1}}{F_n}$ as $n\to\infty$ is $\phi$ if we assume that the limit exists.
How can we prove that the limit does in fact exist?
Is there more than one method?

I do not think this is a duplicate. Please can someone explicitly show how to split the odd and even terms of this ratio sequence into two sequences- one monotonic increasing and one monotonic decreasing- and given that all ratios are between 1 and 2, show that the limit exists and we do not oscillate forever.

Most of the question has been answered.
I have shown that there are two subsequences- one increasing and bounded above by 2 and one decreasing, bounded below by 1. Using the fact that the limit exists, I can show it has value $\phi$. But how can I show the limit is the same for both subsequences?!

REPLY [7 votes]: By Cassini's Identity:
$$\left|\frac{F_{n+1}}{F_{n}}-\frac{F_{n}}{F_{n-1}}\right|=\left|\frac{F_{n+1}F_{n-1}-F_{n}^{2}}{F_{n}F_{n-1}}\right|=\left|\frac{(-1)^{n}}{F_{n}F_{n-1}}\right| \to 0$$
Proof of Cassini's Identity: $$F_{n+1}F_{n-1}-F_{n}^{2}\\ =(F_{n}+F_{n-1})F_{n-1}-F_{n}^{2}\\ =F_{n-1}F_{n}-F_{n}^{2}+F_{n-1}^{2}\\ =-(F_{n}(F_{n}-F_{n-1})-F_{n-1}^{2})\\ =-(F_{n}F_{n-2}-F_{n-1}^{2})$$
You can fill in the rest by induction.