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Carl McTague

A Dynamical Proof That the Ratio of Successive Fibonacci Numbers Approaches the Golden Ratio
In his 1611 essay On the SixCornered Snowflake, Johannes Kepler observed that the ratio of successive Fibonacci numbers $F_n/F_{n1}$ approaches the golden ratio $\tfrac12(1+\sqrt5)$. [Recall that $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n1}$ for $n\ge2$.]
Many proofs have since been found. Here is a quick, dynamical one I thought of which I have been unable to find in the literature.
Proof: Consider the function $f(x)=(1+x)/x$. The sequence of ratios of successive Fibonacci numbers is the orbit of $F_2/F_1=1$ under $f$, i.e. $$ f(F_n/F_{n1})=F_{n+1}/F_n. $$ On the other hand, $f$ has precisely two fixed points, the roots of $x^2x1$, one positive (the golden ratio), the other negative. Since $f'(x)=1/x^2$, $f$ is a decreasing function for $x>0$. And $f(2)=3/2$ while $f(3/2)=5/3$. So $f$ maps the interval $[3/2,2]$ into itself. In fact, $f$ is a contracting mapping on $[3/2,2]$ (with Lipschitz constant $4/9$) since $$ f(x)f(y)=yx/xy\le\tfrac49yx $$ for $x,y\in[3/2,2]$. By the Banach fixed point theorem, then, all orbits in $[3/2,2]$ approach the golden ratio. Although $1$ is not in $[3/2,2]$, $f(1)=2$, so the orbit of $1$ approaches the golden ratio. ∎
This proof can be extended to show that the conclusion holds even if the first two terms $F_1,F_2$ of the Fibonacci sequence are altered, provided their ratio $F_2/F_1$ is not $\tfrac12(1\sqrt5)$ (the negative, unstable fixed point of $f$).
In a sense this explains why the limit is the golden ratio: being a fixed point of $f$ is the defining property of the golden ratio.