I just read the proof of Muirhead Inequality here and I thought it was really cool. But I do have a question, in that proof, nowhere was the fact that ${[a_i]}_{i=1}^n$ majorizes ${[b_i]}_{i=1}^n$ used.
(ie. $a_i$ and $b_i$ are increasing and $$\sum_{i=1}^{k} a_i \ge \sum_{i=1}^{k} b_i \forall i \in [1,n-1] \space \text{and}\space \sum_{i=1}^{n} a_i = \sum_{i=1}^{n} b_i)$$
I mean the first lemma just used that $\sum c_i = 0$ and made no mention of $$\sum_{i=1}^{k} c_i \ge 0 \ \ \ \forall \ \ \ i\in [1,n-1]$$ But surely the majorization condition is really important, right? But the proof doesn't make use of it, so I'm not sure why it is important.
So, essentially I want to know if the proof uses the majorization criteria (and if so, where) and if it doesn't, then why?
Thank you!