TITLE: Proof the golden ratio with the limit of Fibonacci sequence
QUESTION [5 upvotes]: Let $F_n=F_{n-1}+F_{n-2}$ the Fibonacci numbers, and $\phi=\frac{1+\sqrt5}{2}$
The exercise asks me to prove that:  $\lim\limits_{n \to \infty}\frac{F_{n+1}}{F_n}=\phi$.
Sorry as can be proceed??

REPLY [15 votes]: $$F_{n+1}=F_{n}+F_{n-1} \Rightarrow \frac{F_{n+1}}{F_n}=1+\frac{F_{n-1}}{F_n}$$
Let 
$$x_n:= \frac{F_{n+1}}{F_n}$$
Then 
$$x_n=1+\frac{1}{x_{n-1}}$$
You can now prove that $1 \leq x_n \leq 2$ and by induction that
$$ x_1 \leq x_3 \leq x_5 \leq .. \leq x_{2n+1} \leq x_{2n} \leq x_{2n-2} \leq .. \leq x_2$$
From here you get that $x_{2n-1}$ and $x_{2n}$ converge, and you can use their definitions to get their limits. You prove that both limits are the same, which yields the desired result.