I've been asked to show that $x_n \rightarrow L$ as $n \rightarrow \infty$ where $x_n = F_{n+1}/F_{n}$ for $n \in \mathbb{Z}^+$, where $F_n$ denotes the $n^{th}$ Fibonacci number. I am supposed to use this and the fact that $1/x_n \rightarrow 1/L$ as $n \rightarrow \infty$ to prove that $L$ is the golden ratio. I'm not sure how to go about computing this. Should I use the standard $|x_n -L| < \epsilon$ proof? That doesn't seem to get me anywhere when I try it.
Sequences and Series – Fibonacci Sequence and Golden Ratio
convergence-divergencefibonacci-numbersgolden ratiosequences-and-series
Best Answer
Let $R = \frac{F_{n+1}}{F_n}$, and suppose that as $n\to\infty$ this converges to some limit $L$. Then the following is true: $$L = \lim_{n\to\infty}\frac{F_{n+1}}{F_n} = \lim_{n\to\infty} 1 + \frac{F_{n-1}}{F_n} = \lim_{n\to\infty} 1 + \frac{F_{n}}{F_{n+1}} = 1 + \frac1{L}$$
So assuming the above, $$L = 1+\frac{1}{L}\implies L^2 - L - 1 = 0\implies L = \frac{1\pm \sqrt{5}}{2}$$ Clearly, $L$ should be positive, and we are left with $$L = \frac{1+ \sqrt{5}}{2} = \phi$$