Let $H$ and $K$ be normal subgroups of a group $G$. Prove that $H \cap K $ is a normal subgroup of $G$
Showing $H \cap K$ is a subgroup
$$a,b\in H\cap G\Rightarrow ab^{-1} \in H \wedge ab^{-1}G \Rightarrow ab^{-1}\in H \cap G$$
Having trouble with $H\cap K\unlhd G$
trying to use the following definitions
$N \unlhd G$
$$\begin{aligned} a^-1Na=N, \text{ forall } a\in G \\ aNa^{-1}=N, \text{ forall } a\in G \end{aligned} $$
since $g\in H \cap K$
$$ \begin{aligned} gHg^{-1}=H \\gKg^{-1}=K \end{aligned} $$
I did read a previous question this and could not convince myself of the proof.