Continued Fraction Expansion of Golden Mean

Theorem
The golden mean has the simplest possible continued fraction expansion, namely $\sqbrk {1, 1, 1, 1, \ldots}$:
 * $\phi = 1 + \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots} } }$

Proof
Let:
 * $x = 1 + \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots} } }$

Then:

The result follows from Golden Mean as Root of Quadratic.