I have been considering a few proofs of this theorem, and I noticed that a few of them (for example Proof 1, and Proof 2) prove the theorem first for homogeneous symmetric polynomials and then generalise it. However I have seen other proofs which don't seem to require this step (e.g. Proof 3) and it seems to me superfluous.
Could anyone explain why so many proofs include this step and/or if it is necessary?