Proof. Denote the intersection of subgroups of G as
\begin{align*}
K = \cap_{i \in I} H_i
\end{align*}Intersection of subgroups of G is a subgroup of G
Intersection of subgroups of G is a subgroup of G.
Proof. Denote the intersection of subgroups of G as
where for all i \in I, H_i are subgroups of G.
First, we will check that K is non-empty, i.e., it consists of the identity element. Now, each subgroup of G contains an identity element. Intersecting all the subgroups has at least the identity element. Therefore, K contains the identity element, and so K is non-empty.
Secondly, we want to know that if x,y \in K, then xy^{-1} \in K. Since x,y \in K, we have that x,y \in H_i \ \forall i \in I. We know that H_i is a subgroup of G for all i \in I. So we get that xy^{-1} \in H_i for all i \in I. So xy^{-1} \in K.
Proof. Denote the intersection of subgroups of G as
