Remark 1

Let $p$ be an integer such that $N_p= N_{p+1}$; the proof given above shows that $N_{\infty}\cap I_p= 0$, and the restriction of $u$ to $I_p$ is injective. Likewise, let $q$ be an integer such that $I_q= I_{q+1}$; then we have $N_q+ I_{\infty}= M$, and the endomorphism of $M/N_q$ deduced from $u$ by passing to the quotient is surjective.