Equivalence of Definitions of Generated Normal Subgroup

Theorem
Let $G$ be a group.

Let $S \subset G$ be a subset.

$(1)$ is equivalent to $(3)$
Let $H$ be the smallest normal subgroup containing $S$.

Let $\mathbb S$ be the set of normal subgroups containing $S$.

To show the equivalence of the two definitions, we need to show that $H = \bigcap \mathbb S$.

Since $H$ is a normal subgroup containing $S$:
 * $H \in \mathbb S$

By Intersection is Subset:
 * $\bigcap \mathbb S \subseteq H$

On the other hand, by Intersection of Normal Subgroups is Normal:
 * $\bigcap \mathbb S$ is a normal subgroup containing $S$.

Since $H$ be the smallest normal subgroup containing $S$:
 * $H \subseteq \bigcap \mathbb S$

By definition of set equality:
 * $H = \bigcap \mathbb S$

Hence the result.