Show that the union of a nonempty directed family of subgroup of a group G is a subgroup of G.

Proof so far:

Let G be a group, and suppose that be a directed family of subgroups of G.

Then is the union.

Since it is nonempty, the subgroup of {e}, act as the identity of this subgroup.

Now, suppose that x and y are elements of this union, then there exist i,j in A such that and . And since this family is directed, there exist some k in A such that and . Implies that both x,y are in And since the union is consist of all subgroups, that means is a subgroup, implies that , thus proves the claim.

Is this right? Thank you.