Four tuple/Modulus of differences/Descent/Exercise

We consider the mapping

$\Psi \colon \mathbb {N} ^{4}\longrightarrow \mathbb {N} ^{4}$

which assigns to a four tuple ${}(a,b,c,d)$ the four tuple

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

Show that for any initial value ${}(a,b,c,d)$, after finitely many iterations this map reaches the zero tuple.

Create a solution