I have some questions about the proof of a Corollary to the Whitney Embedding Theorem from John Lee's Smooth Manifolds.
In the proof below, why is the product map $G=f\times F$ a smooth embedding? I can see that it is an injective smooth immersion since $F$ is, but how can I show that this is also a topological embedding, i.e. a hoemomorphism onto its image $G(M)\subset \mathbb{R}^N \times \mathbb{R}^{2n+1}$ in the subspace topology?
Also, how can I show that if a vector $v_{N+2n+1}$ is arbitrarily close to $e_{N+2n+1}=(0,\dots,0,1)$ then the orthogonal projection map $\pi_{v_{N+2n+1}}$ is arbitrarily close to $\pi_{e_{N+2n+1}}$?
I would greatly appreciate any help with these questions.
A bit late and nothing that hasn’t been said in the comments but maybe a worthwhile thought due to its explicitness:
The map $G=f\times F: M\to \mathbb{R}^N\times\mathbb{R}^{2n+1}$ is a homeomorphism as the composition of the projection $\mathbb{R}^N\times\mathbb{R}^{2n+1} \to \mathbb{R}^{2n+1}$ and the inverse map $F^{-1}:F(M)\to M$ induces the inverse map of $G$ and is continuous. So indeed, this part of the proof does not depend on the compactness of $M$.
The second part relies on compactness to give a uniform approximation.