Neighborhood Retraction of Boundary

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.

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I'm curious - why would you use Zorn's lemma instead of a partition of unity? –  Neal Aug 22 '12 at 21:21
Because I cannot figure out how to make the argument work with partitions of unity. If someone can give me a hint on how to make partitions of unity work in this case, then I will gladly amend my approach to the theorem. –  Joshua G Aug 22 '12 at 21:47