Proof of Correctness Request for Greedy Algorithm that solves “The Weight Job Scheduling” problem

Today, in my self-lead studies, I found out about greedy algorithms, more specifically, a greedy approach to solve The Weighted Job Scheduling Problem.

I understand how the solution is implemented but, I’d love to see a proof of correctness for this solution (i.e. partial correctness and termination). If anyone can help me understand why this solution is correct mathematically in a general form, that’d really great!