Ratio of Consecutive Fibonacci Numbers

Theorem
For $n \in \N$, let $f_n$ be the $n$th Fibonacci number.

Then:
 * $\displaystyle \lim_{n \to \infty} \frac {f_{n + 1}} {f_n} = \phi$

where $\phi = \dfrac {1 + \sqrt 5} 2$ is the golden mean.

Proof
Denote:
 * $\phi = \dfrac {1 + \sqrt 5} 2$, $\hat \phi = \dfrac {1 - \sqrt 5} 2$

and:
 * $\alpha = \dfrac {\phi} {\hat \phi} = - \dfrac {3 + \sqrt 5} {2}$

From the Euler-Binet Formula:


 * $f_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$

Let $n \ge 1$.

It immediately follows that:

From the definition of $\alpha$:
 * $|\alpha| > 1$

Therefore:
 * $\displaystyle \lim_{n \to \infty} \frac {f_{n + 1}} {f_n} = \lim_{n \to \infty}\ \phi + \dfrac {\sqrt 5} {\alpha^n - 1} = \phi$