I am reading the proof of the Whitney Embedding Theorem from John Lee's Introduction to Smooth Manifolds. However, I cannot figure out some statements from the proof. The proof relies on Lemma 6.14 that states if $M$ is a smooth $n$-manifold with or without boundary and $M$ admits a smooth embedding into $\mathbb{R}^N$ for some $N$, then it admits a proper smooth embedding into $\mathbb{R}^{2n+1}$.
The first part of the proof proves the theorem in the case $M$ is compact. Then according to the errata of the book, this argument applies to the case when $M$ is an arbitrary compact subset of a larger manifold $\tilde{M}$ with or without boundary, by covering $M$ with finitely many coordinate balls or half-balls for $\tilde{M}$. The result is a smooth injective map $F:M\to \mathbb{R}^{nm+m}$ whose differential is injective at each point. [This is needed in the ensuing argument for the noncompact case, because the sets $E_i$ might not be regular domains when $\partial M \neq 0$.]
However, in the second part of the proof, I cannot see how lemma 6.14 applies to the compact sets $E_i$. So the first part shows that for each $i$ there is a smooth injective map of $E_i$ into some Euclidean space whose differential is injective eat each point. But $E_i$ here, in the case $\partial M \neq 0$, is only a compact subset of $M$, so without a reference as to it being a codimension-0 submanifold of $M$, the hypothesis of a smooth $n$-manifold with or without boundary is not satisfied for $E_i$. Indeed, the definition of smooth embedding requires the domain to be a smooth manifold but here we only have that $E_i$ is a compact subset of a smooth manifold. So how do we ensure an embedding $\varphi: E_i \to \mathbb{R}^{2n+1}$?
Finally, in the $F$ constructed, how do we know that $F$ is proper because $f$ is?
I would greatly appreciate some help here.
Best Answer
Let me answer your last question.
Lemma. Suppose that $X, Y, Z$ are Hausdorff topological spaces, $f:X\to Y, g: X\to Z$ are continuous and $f$ is proper. Then $F=f\times g: X\to Y\times Z$ is also proper.
Proof. Consider a compact $K\subset Y\times Z$; I'll prove that its preimage is compact. Let $A, B$ denote the projections of $K$ to $Y, Z$ respectively; both are compact. Then $F^{-1}(K)\subset f^{-1}(A)\cap g^{-1}(B)$, a closed subset of the compact $f^{-1}(A)$. Hence $F^{-1}(K)$ is compact. qed