# Upper and Lower Bound of Fibonacci Number

## Theorem

For all $n \in \N_{> 0}$:

$\phi^{n - 2} \le F_n \le \phi^{n - 1}$

where:

$F_n$ is the $n$th Fibonacci number
$\phi$ is the golden section: $\phi = \dfrac {1 + \sqrt 5} 2$

## Proof

$F_n \ge \phi^{n - 2}$
$F_n \le \phi^{n - 1}$

$\blacksquare$