Ratio of Consecutive Fibonacci Numbers/Proof 1

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

Then:

Recall 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 $\alpha$:
 * $\size \alpha > 1$

Therefore: