Question

Prove that intersection of two subgroups is also a subgroup

Answer 1

You've done almost all of it already:

Let G be a group, H1,H2 subgroups of G, H=H1∩H2. Then

H is not empty: If e is the neutral element of G, then e∈H1 and e∈H2, because these are subgroups. Hence also e∈H and H≠∅.

If a,b∈H, then ab−1∈H: Indeed, a,b∈H implies a,b∈H1 as H⊆H1, hence ab−1∈H1 because H1 is a subgroup. Similarly, ab−1∈H2 and hence ab−1∈H.

You should know the subgroup criterion: A subset H of a group is a subgroup iff H≠∅ and a,b∈H implies ab−1∈H. Hence we have just shown that H is a subgroup of G.