I am trying to study chapter 6 of Lee's Introduction to Smooth Manifolds.
In a remark after the Whitney Approximation Theorem, Lee stated that this theorem do not hold because it might not be possible to find a smooth map that agree with $F$ on $A$.
I am trying to chase back to his proof on the Whitney Approximation Theorem, but I am not sure when the empty boundary assumption comes into play.
Part of the proof goes like this:
By the Whitney embedding theorem, we can assume $M$ is a properly embedded submanifold of $\mathbb{R}^n$. Then we can find a tubular neighborhood $U$ of $M$ in $\mathbb{R}^n$ and a smooth retraction $r: U \rightarrow M$.
Define $\delta(x) = \sup\{\varepsilon \leq 1: B_{\varepsilon}(x) \subseteq U\}$ for any $x \in M$ and claims that $\delta: M \rightarrow \mathbb{R}^+$ is continuous and define $\tilde{\delta} = \delta \circ F: N \rightarrow \mathbb{R}^+$. Then by the Whitney Approximation Theorem for Functions there exists a smooth function $\tilde{F}: N \rightarrow \mathbb{R}^n$ that is $\tilde{\delta}$-close to $F$ and is equal to $F$ on $A$.
My question: Both Whitney Embedding Theorem and Whitney Approximation Theorem for Functions hold for the case of manifolds with boundaries, so which of the preceding steps might fail in the case if the co-domain $M$ actually has non-empty boundary and why?
Thank you very much.
Here's a screenshot of the theorem and the proof:
