I'd like to check the veracity of my proof. I've seen several proofs using different methods (some I'm allowed to use with lots of element-pushing and others using ideas I'm not allowed), but none explicitly like mine.
First, a definition:
Definition: A group $G$ is solvable if there is a chain of subgroups $$ 1 = G_0 \triangleleft \dots \triangleleft G_k =G$$ such that $G_{i+1}/G_{i}$ is abelian for $i = 0, 1, \dots, k-1.$
Problem: Quotient groups of a solvable group are solvable.
Proof: Let $\overline{G}$ be the quotient of a solvable group $G$ by some normal subgroup $N$ of $G$. G is solvable so there is a chain:
$$ 1 = G_0 \triangleleft \dots \triangleleft G_n =G$$
such that $G_{i+1}/G_{i}$ is abelian for $i = 0, 1, \dots, n-1$. By the lattice isomorphism theorem, $$G_{i} \triangleleft G_{i+1} \iff \overline{G_{i}} \triangleleft \overline{G_{i+1}}$$ so there is a chain: $$ \overline{1} = \overline{G_{0}} \triangleleft \dots \triangleleft \overline{G_{n}} = \overline{G}$$ for the quotient group. Now, by the 3rd isomorphism theorem, we have
$$ \overline{G_{i+1}} / \overline{G_{i}} = (G_{i+1} / N)/(G_{i} / N) \cong G_{i+1} / G_{i}$$ which are abelian. $\Box$
The lattice isomorphism theorem gives you a correspondence between subgroups of $G/N$ and subgroups of $G$ which contain $N$. So your proof isn't quite right if any of the $G_i$ fail to contain $N$.