Q:

Proof from John Lee on the equivalence of topological boundary and manifold boundary for regular domains

I am having difficulty following the conclusion of the proof below from John Lee's Introduction to Smooth Manifolds. Here a regular domain in $M$ is a properly embedded codimension-$0$ submanifolds with boundary. 
Here $F$ is the inclusion map from $D \hookrightarrow M$, which is a smooth embedding.
I understand the proof before the final sentence. But how do we conclude from here that every neighborhood of $p$ intersects both $D$ and $M\backslash D$? 

A:

I found this proof confusing, too. Here is my understanding: we have that $D$ is an embedded submanifold of $M.$ Take any $p\in \partial D$. Then, there is a chart $\textit{in M},\ $ say, $(V,(x^1,\cdots,x^n))$ about $p$ such that $(D\cap V,(x^1,\cdots,x^k))$ is a boundary (slice) chart for $D$ about $p$. Thus, $q\in D\cap V\Rightarrow x^k(q)\ge 0$ But $\text{dim}\ D=\text{dim}\ M\Rightarrow k=n$, and so $x^n\ge 0.$ Now, $M$ is a manifold without boundary, which means that there must be a point $q\in V$  such that $x^n(q)<0$, (because $(V,(x^1,\cdots,x^n))$ is a chart about $p$ in $M$), which in turn implies that $q\notin D$ (because $D\cap V$ has all $x^n\ge 0$). Therefore, $V$ contains points in $D$ and in $M\setminus D$.

