# Equivalence of Definitions of Upper Wythoff Sequence

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## 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.

$\blacksquare$