# Equivalence of Definitions of Upper Wythoff Sequence

## Theorem

The following definitions of the upper Wythoff sequence are equivalent:

### Definition 1

The upper Wythoff sequence is the complementary Beatty sequence on the golden section $\phi$.

### Definition 2

The upper Wythoff sequence is the Beatty sequence on the square $\phi^2$ of the golden section $\phi$.

## Proof

From Beatty's Theorem, the Beatty sequences $\mathcal B_r$ and $\mathcal B_s$ are complementary if and only if:

$\dfrac 1 r + \dfrac 1 s = 1$

It remains to be demonstrated that this holds for $r = \phi$ and $s = \phi^2$.

Thus:

 $\ds \dfrac 1 \phi + \dfrac 1 {\phi^2}$ $=$ $\ds \dfrac {\phi + 1} {\phi^2}$ $\ds$ $=$ $\ds \dfrac {\phi^2} {\phi^2}$ Square of Golden Mean equals One plus Golden Mean $\ds$ $=$ $\ds 1$

Hence the result.

$\blacksquare$