Group Generated by Normal Intersection is Normal

Theorem
Let $I$ be an indexing set, and $\set {N_i: i \in I}$ be a set of normal subgroups of the group $G$.

Then $\family {N_i: i \in I}$ is a normal subgroup of $G$.

Proof
By definition, $\family {N_i: i \in I}$ is the intersection of all the subgroups of $G$ which contain every $N_i$.

For each $H \le G$, the conjugate $g H g^{-1}$ contains each $g N_i g^{-1}$.

Since each $N_i \lhd G$, it follows that:
 * $N_i \subseteq g H g^{-1}$

Thus it follows that:
 * $\family {N_i: i \in I} \lhd G$