Direct Product of Group: Isomorphic Subgroups

Question

Show that a Direct product of groups, $G \times H$, contains subgroups isomorphic to both $G$ and $H$.

This seems like a simple questions but I can't wrap my head around it. I've already proved that $G \times H$ is a group

Answer 1

Hint: $$\phi: G \longrightarrow G \times H, \,\,\ g \mapsto (g,e_H).$$ Show that $\phi$ is an injective group homomorphism. Similarly for $H$.