We keep the notations from Parts II and III. We denote $r _ { k } = b - A x _ { k }$, $e _ { k } = x _ { k } - \tilde { x }$, and note that $r _ { k } = - A e _ { k }$. We denote by $m$ the smallest integer $k$ such that $H _ { k + 1 } = H _ { k }$.
Show that $e _ { k } \neq 0$ for $k \in \{ 0 , \ldots , m - 1 \}$, and that $e _ { k } = 0$ for $k \geq m$.