I need to prove that $H$ is a maximal normal subgroup of $G$ if and only if $G/H$ is simple.
My proof to the $(\Rightarrow)$ direction seems too much trivial:
Let us assume there exist $A$ so that $A/H\lhd G/H$. Then by definion, $H$ must be normal in $A$. Because $H$ is maximal, we get $H=A$ and therefore $A/H={1}$
Is it correct?
Update:
Now I see that I need to prove that not only $A\lhd H$ but also $A\lhd G$. Assumig I have proven that, is the proof correct?