surjection of complete noetherian local rings -reference help

Question

I am looking Lemma 1.1: $\widehat{C}$ is the category of complete local noetherian rings.

Is there a reference to the last two lines of the above argument ? i.e. : associated graded rings being surjective implying original ring map is surjective?

It says the reference is to Bourbaki Chapter III. However,I couldn't find the citation [2, No.8, Theorem 1]. What exactly does this mean?

Answer 1

See Chapter 10: Completions, Lemma 10.23 in Atiyah-MacDonald: Commutative algebra book. It follows that if the induced map of associated graded rings is surjective, then the original map is also surjective.