Solution: Let $N_1$ and $N_2$ be two normal subgroups of $G$. We need to show that $N_1 \cap N_2$ is normal subgroup of $G$. We know that $N \coloneqq N_1 \cap N_2$ is a subgroup. To show, it is normal subgroup, take $g \in G$. We need to show that $gNg^{-1} \subseteq N$. For any $n \in N = N_1 \cap N_2$, we know that $gng^{-1} \in N_1 $ and $gng^{-1} \in N_2$, therefore, $gng^{-1} \in N$. Thus, $N$ is a normal subgroup of $G$.