Equivalence of Definitions of Upper Wythoff Sequence

Theorem
The following definitions of the upper Wythoff sequence are equivalent:

Proof
From Beatty's Theorem, the Beatty sequences $\mathcal B_r$ and $\mathcal B_s$ are complementary :
 * $\dfrac 1 r + \dfrac 1 s = 1$

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

Thus:

Hence the result.