Definition 1: $f \in C^{\infty}(M,N)$ is called immersion iff $\forall x \in M: \operatorname{Ker} d_x f = 0$.
Definition 2: $f \in C^{\infty}(M,N)$ is called embedding iff it is immersion and topological embedding (homeomorphism on image).
I have to prove following:
- Every immersion is a local embedding
- Image of embedding is a submanifold
- Embedding is a diffeomorphism between $M$ and $f(M)$.
This theorem is left without a proof because it is called "easy". But i would like to know the proof, maybe i can read it somewhere?
What i understand is that (3) easily follows from inverse function theorem. Also, (1) easily follows from (3).
And one more thing disturbs me: $M$ can have non-empty boundary. Does that call some difficulties?