Show that the intersection of normal subgroups is normal.
Let $H_1$ and $H_2$ be normal in $G$, meaning $\forall a \in G$, $aH_1 = H_1a$ and $aH_2 = H_2a$. We show that $a (H_1 \cap H_2) = (H_1 \cap H_2)a$.
Now notice $aH_1 \cap aH_2 = H_1a \cap H_2 a$, so we are done.
My book does something completely off and they showed that $(H_1 \cap H_2)$ is actually a subgroup first, then they used conjugation to prove normality. I thought showing left and right cosets coinciding is enough?