Proof. Applying Algebra, Lemma 10.99.1 we see that $u$ is injective and that $N/u(M)$ is flat over $A$. Then $u$ is $A$-universally injective (Algebra, Lemma 10.39.12) and $N$ is $A$-flat (Algebra, Lemma 10.39.13). Since $B$ is Noetherian in this case we see that $N$ is of finite presentation. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).