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