I wanted to prove $\partial \bar N\subset \partial N$ in a different way, by showing if $x \in \partial \overline{N}$ then $x \in \partial N$. This I wanted to show by showing that if $U$ is an nbhood of $x$ then it contains
(i) a point of $N$ and
(ii) a point of $N^c$.
Since $U$ is an nbhood of $x$ it contains a point of $\overline{N}$ and a point of $\overline{N}^c$. We have $N \subset \overline{N}$ hence $\overline{N}^c \subset N^c$ hence (ii) is clear. What I'm stuck with is (i). I tried the following: $U$ contains a point $y$ of $\overline{N} = \partial N \cup \mathrm{int}N$. If $y \in \mathrm{int}N$ then we're done. If $y$ in $\partial N$ then either $y \in N$ or $y\in N^c$. If $y \in N$ we're done.
But if $y \in N^c$ then... nothing. So it looks as if $U$ could be a subset of $N^c$. Is it really not possible to prove it like this or do I just not see how? Thanks for help.