USACO Bronze 2021 December - Air Cownditioning

Hello all, to anyone who is reading this post. Thank you for taking your time to help me. I looked at Air Cownditioning’s O(N) solution in the official USACO analysis. But I didn’t quite understand it. To check it out, you can go to this link: Contest Results, and scroll to the end. I didn’t understand the claim and the proof. Can anyone explain them to me? Thanks.