Intersection of subgroups is a subgroup

Any intersection of subgroups of a fixed group is again a subgroup

Intersection of subgroups is a subgroup

Proposition (Intersection of subgroups). Let $G$ be a group and let $\{H_i\}_{i\in I}$ be a family of subgroups of $G$ . Then the set-theoretic intersection

H \;=\; \bigcap_{i\in I} H_i

is a subgroup of $G$ .

Context. This implies there is a smallest subgroup of $G$ containing any given subset (the intersection of all subgroups containing it), which underlies the notion of a generated subgroup.