I'd like some feedback from experts in group theory.
Let $N \trianglelefteq G$, $G$ solvable. Then $G/N$ is solvable.
My proof: Take canonical homomorphism $G \rightarrow G/N$ ($x \rightarrow xN$) and let abelian tower $\{e\} = G_0 \trianglelefteq \ldots \trianglelefteq G_m = G $. Image of normal group under surjective homomorphism is normal, thus we have a tower $N = f(G_0) \trianglelefteq \ldots \trianglelefteq f(G_m) = G/N$. By one of isomorphism theorems, $G_{i+1} / G_i\cong G_{i+1} / N / G_i / N$, thus tower is abelian.
Where did I go wrong? And if I am correct, why is "standard proof" using more complicating structures such as $G_iN/N$?
Edit: I know that $G_i/N$ only makes sense for $N \subset G_i$. But for those $i$ such that $G_i \subset N$, canonical homomorphism gives $f(G_i)=N$ (collapse) so the whole thing still stands, i.e. we have this tower
$$N \trianglelefteq \cdots \trianglelefteq N \trianglelefteq G_{i+1}/N\trianglelefteq \cdots \trianglelefteq G_m/N = G/N$$