I'm trying to understand a proof in the subject of differential manifolds. In the book im reading, there is a theorem that states: "Let $M$ be a differential manifold with a local finite atlas, then there exist $N \in \Bbb N$ and an $embedding$ of M into $\Bbb R^N$ "
the proof structure goes as follows:
- First they defining a map smooth map $F:M \to \Bbb R^N$. Showing this map $F$ is injective, and that $F^{-1}:F(M) \to M$ has a local smooth extension, which means that $F$ is homeomorphic to its image $F(M)$.
- Secondly, showing the differential $dF_p:T_pM \to T_{F(p)}\Bbb R^N$ is injective for every p, so F is immersive.
- showing that $F(M)$ is an m-dim manifold in $\Bbb R^N$
- Concluding that $F$ is a diffeomorphism $M \to F(M)$.
Now, my question-
From what I learned, an embedding $f:M \to N$, means that $f(M)$ can be given with the structure of embedded submanifold of N (i.e f(M) is submanifold, which have the induced topology of N, such that the inclusion $i:f(M) \to N$ is smooth and immersion.)
So, How the proof provided above telling anything about the inclusion $i:F(M) \to \Bbb R^N$? And how can I conclude this inclusion is immersive at any p? And also, why do I need $F$ to be diffeomorphism in order to show its an embedding?
thanks a lot!