Group Generated by Normal Intersection is Normal

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

Then $\left \langle {N_i: i \in I} \right \rangle$ is a normal subgroup of $G$.

Proof
By definition, $\left \langle {N_i: i \in I} \right \rangle$ 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 $N_i \triangleleft G$, it follows that $N_i \subseteq g H g^{-1}$.

Thus it follows that $\left \langle {N_i: i \in I} \right \rangle \triangleleft G$.