Four tuple/Modulus of differences/Q/Descent/Exercise

We consider the mapping

$\Psi \colon \mathbb {Q} _{\geq 0}^{4}\longrightarrow \mathbb {Q} _{\geq 0}^{4}$

that assigns to a four-tuple of nonnegative rational numbers ${}(a,b,c,d)$ the four-tuple

$(\vert {b-a}\vert ,\vert {c-b}\vert ,\vert {d-c}\vert ,\vert {a-d}\vert )$

Show that after finitely many iterations, this mapping yields the zero-tuple.

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