I have posted my proof of the Pigeonhole Principle.
The Attempt at a Solution
Basically I am curious if this is an acceptable proof of the Pigeonhole Principle? I ask because both my professor and our textbook complete it differently, both of which make use of the restriction of a function. I choose this form for two simple reasons, I am having a hard time understanding why they are employing the restriction of a function and why they are employing the function g, but also this seems a lot simpler. I tend to make these things more difficult than they ought to be, but in mathematics nothing is arbitrary except the objects we place in sets. I have included both my version and the version my book used. My professor gave a proof similar to the one in the book. Thanks in advance. The first screenshot is my proof.