Intersection of subgroups is a subgroup

Statement

Let $G$ be a group and let $(H_i)_{i \in I}$ be a family of subgroups of $G$, indexed by a nonempty set $I$. Then the intersection $\bigcap_{i \in I} H_i$ is a subgroup of $G$.

Remarks

The identity is in each $H_i$, and if $a, b$ are in each $H_i$, then $ab^{-1}$ is in each $H_i$. The union of two subgroups is in general not a subgroup: in $(\mathbb{Z}, +)$, $2 \in 2\mathbb{Z}$ and $3 \in 3\mathbb{Z}$, but $5$ is in neither. The union of two subgroups $H$ and $K$ is a subgroup only when $H \subseteq K$ or $K \subseteq H$.

Explanations

L1 human

If several subgroups each pass the subgroup test, then their common part also passes it. For example, $2\mathbb{Z} \cap 3\mathbb{Z} = 6\mathbb{Z}$ in the integers. This fact lets us define the smallest subgroup that contains a given subset.