Sard's theorem and Whitney approximation

Question

I'm confused about John Lee's proof of Whitney approximation theorem in Introduction to Smooth Manifolds. It claims that the theorem is an application of Sard's theorem, but I didn't find where it was applied. Also, he didn't prove the case that $F$ is not smooth on some closed set, and I've got no clue to utilizing the pattern (partition of unity) he showed. link

Answer 1

You're right. In that proof, the author did not use Sard Theorem. The theorem that you refer to is called Whitney Approximation Theorem for Functions, which means that the continuous function on manifold that we want to approximate has Euclidean space as the target space. As pointed out by Ted Shifrin, we can always take $A=\emptyset$ if there's no such closed subset $A$.

The claim that Sard Theorem is used to prove Whitney approximation theorem is not about Whitney Approximation Theorem for Functions above, but about The Whitney Approximation Theorem (Theorem 10.21 proved later in the book) which involves function between manifolds, and the first step to prove this is to embed our target space (a manifold) into some Euclidean space using Whitney Embedding Theorem, which is a corollary of Sard Theorem.