Proposition 1: Let $0\to M' \xrightarrow{f} M \xrightarrow{g} M''\to 0$ be an exact sequence of $A$-modules. Then:
i) $M$ is Noetherian iff $M'$ and $M''$ are Noetherian.
ii) $M$ is Artinian iff $M'$ and $M''$ are Artinian.
Proposition 2: Let $A$ be a Noetherian (Artinian), $I$ an ideal of $A$. Then $A/I$ is a Noetherian (Artinian) ring.
My question is the following: I know the proof of Proposition 1. How to use it in order to prove proposition 2?
I was trying to construct some examples of exact sequences but I failed to do it.
The exact sequence you want is $$0\to I\to A\to A/I\to 0$$ $A$ is Noetherian or Artinian, hence so is $A/I$.