differential topology – Proof of collar neighborhood theorem

I’m trying to understand the proof of the collar neighborhood theorem given in the following document:

http://www.math.toronto.edu/vtk/1300Fall2015/lecture-nov2.pdf

At the end of the proof it says that we can argue by contradiction using the compactness of $partial M$ to construct the desired diffeomorphism, but I don’t see how too do it.

Could anybody help me?

Thanks in advance.