Abelian implies all subgroups normal

Proposition (Subgroups of abelian groups are normal). Let $G$ be an abelian group and let $H\le G$ be a subgroup . Then $H$ is a normal subgroup of $G$.

Context. Normality is a conjugation-invariance condition. In an abelian group, conjugation is trivial: $ghg^{-1}=h$ for all $g,h$.

Proof sketch. Fix $g\in G$ and $h\in H$. Since $G$ is abelian, $ghg^{-1}=h\in H$. Hence $gHg^{-1}\subseteq H$ for all $g$, so $H\lhd G$.