In Rudin, we are given this corollary, 7.27 to the Stone Weierstrass Theorem:
Say instead I replaced $|x|$ in the corollary with a continuous function such that $g(0)=0$, how would the proof change. I honestly don't see any specific changes, although I really feel as though I'm missing something important. Also I'm also confused by Rudin's proof in the first place, by the last line in particular. What does he mean: the polynomials $P_n(x)=P_n^*(x)-P_n^*(0) (n=1,2,3..)$ have the desired properties? Any help would be much appreciated. Thank you.