I have tried looking through various books and websites and could not find a proof of the existence of partitions of unity for smooth manifolds with boundary. I would like a proof or a reference to one.
2026-03-27 09:48:08.1774604888
How to prove the existence of partitions of unity for smooth manifolds with boundary?
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The following is in German, if you can read it:
https://people.math.ethz.ch/~blatter/Analysis_14.pdf
The first page of chapter 14 is 223. Scroll down to pp. 255–261. There a partition of unity is constructed for domains $B\subset{\mathbb R}^n$ having a smooth boundary. Maybe you can translate the setup to your situation.