Given a finite solvable group G, my teacher just used the fact that it has a composition series. I don't see why it's immediate, or even necessarily true.
If $(G_0={e} ,G_1,...,G_n=G)$ is the normal series because of which G is solvable, we have by definition that $\forall 0\leq i\leq n-1$ that $G_{i+1}/G_i$ is abelian, meaning every subgroup is normal. But it doesn't mean that $G_{i+1}/G_i$ is simple as we'd like.
Notation might be weird (I think it's mostly noted the other way around) but that's how it was defined in my class, so excuse me if it's confusing.