If $\displaystyle G$ is a finite group and $\displaystyle H$ is a normal subgroup of $\displaystyle G$. Prove that there is a composition series of $\displaystyle G$, one of whose terms is $\displaystyle H$.

Recall that every finite group has a composition series.

Definition: A composition series for $\displaystyle G$ is a series of subgroups of $\displaystyle G$: $\displaystyle {1}$ is a normal subgroup of $\displaystyle N_0$ is a normal subgroup of $\displaystyle N_1$ is a normal subgroup of $\displaystyle ...$ is a normal subgroup of $\displaystyle N_k = G$ such that $\displaystyle N_i$ is a normal subgroup of $\displaystyle N_{i+1}$ and $\displaystyle N_{i+1}/N_i$ is simple.

Proof:

$\displaystyle G$ is a finite group, so $\displaystyle G$ has a composition series. And $\displaystyle H$ is a normal subgroup of $\displaystyle G$, by the definition of a composition series, $\displaystyle H$ must be a term of the series.

Does this complete the proof? If not, please help. Thank you.