Let $N$ be a subgroup of a group $G$. By definition, an element $g\in G$ is said to normalize $N$ if $gNg^{-1}=N$. I am trying to show every element of $N$ normalizes $N$.

In other words, I want to show for all $n\in N$, $nNn^{-1}=N$.

My attempt:

Fix $n\in N$ and let $x\in nNn^{-1}$. Then $x=nhn^{-1}$ for some $h\in N$. Since $N$ is a group, $x\in N$. So $nNn^{-1}\subset N$.
I am having a trouble showing the other inclusion. Let $y\in N$. I want to show that $y\in nNn^{-1}$ in order to show $N\subset nNn^{-1}$. Can someone help me figure this out?