Sequence of Wythoff Pairs contains all Positive Integers exactly Once Each

Theorem
Consider the sequence of Wythoff pairs arranged in sequential order:
 * $\left({0, 0}\right), \left({1, 2}\right), \left({3, 5}\right), \left({4, 7}\right), \left({6, 10}\right), \left({8, 13}\right), \ldots$

Apart from the first Wythoff pair $\left({0, 0}\right)$, every positive integer appears in this sequence exactly once.

Proof
By definition, the $n$th Wythoff pair is $\left({\left\lfloor {n \phi}\right\rfloor, \left\lfloor {n \phi^2}\right\rfloor}\right)$.

Thus the coordinates of the sequence of Wythoff pairs are the terms of the lower and upper Wythoff sequences.

By definition:
 * the lower Wythoff sequences is the Beatty sequence on the golden section $\phi$.


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

Also by definition, the complementary Beatty sequence on $x$ is the integer sequence formed by the integers which are missing from $\mathcal B_x$.

Hence the result.