Showing that a subnormal series for a finite group $G$ can be made into a composition series for $G$

Question

Suppose that $\{e\} < G_1 < G_2 < \cdots < G_n = G$ is a subnormal series, thus for all $i$, we have $G_i$ is normal in $G_{i+1}$. How can I show that this can be "refined" to a composition series?

By composition series I mean a subnormal series where, for each $i, G_{i+1}/G_i$ is a nontrivial, simple group.

Answer 1

Suppose $G_{n+1}/G_n$ is not simple, then we have $NG_n$ an intermediate normal subgroup. Rinse and repeat.