Here is the problem: If $M$ is a manifold with boundary, then find a retraction $r:U \rightarrow \partial M$ where $U$ is a neighborhood of $\partial M$.
I realize that the collar neighborhood theorem essentially provides the desired map, but I am actually using this result to prove the aforementioned theorem. My thought on how to prove the theorem is to
1) Show local existence.
2) Show local extendability of these retractions.
3) Use Zorn's Lemma to piece together a neighborhood retraction of the boundary.
The steps with local existence and with Zorn's Lemma are pretty much trivial, so the major difficulty involves local extendability. If anybody could help with this step it would be appreciated.