Showing that a diffeomorphism preserves the boundary

Question

I am supposed to use the fact presented below to show that the diffeomorphism $F$ in the theorem 2.18 preserves the boundary. I found a way to prove it but it does not use the fact below. I am curious how I can use the theorem 1.46 to show that $F$ preserves the boundary. Could anyone please help me?

Answer 1

Let $p \in \partial M$ and consider some chart $\varphi:U\longrightarrow\mathbb H^n$ such that $U\subset M$ with $p\in U$. Now, take some chart $\psi :V\longrightarrow\mathbb H^n$ such that $V\subset N$ with $F(p)\in V$.

We assume without loss of generality that $F(U)\subset V$, otherwise just take $U'\subset U$, still satisfying $p\in U$, with $F(U')\subset V$.

The composition $g=\psi\circ F\circ\varphi^{-1}:\varphi(U)\longrightarrow \psi(F(U))$ is a smooth map between subsets of $\mathbb H^n$, so it is a 'classical' smooth map between subsets of Euclidean space. Moreover, it is invertible and $g^{-1}$ is also smooth, so $g$ is a diffeomorphism between $\varphi(U)$ and $\psi(F(U))$.

Lemma: Let $A,B\subset \mathbb H^n$ be open and $x \in A\cap\partial \mathbb H^n$. If $f:A\longrightarrow B$ is a homeomorphism, then $f(x)\in B\cap\partial \mathbb H^n$.

After showing the lemma, conclude that $g(\varphi (p))=\psi(F(p))\in \partial \mathbb H^n$.

Now apply Theorem $1.46$ to conclude that any chart $(V',\psi')$ on $N$ with $F(p)\in V'$ satisfies $\psi'(F(p))\in \partial \mathbb H^n$. In other words, $F(p)\in \partial N$.