On a theorem of Jackson

We prove that
$$ \inf_{L_n\in Z_n}\sup_\omega\,^*\sup_{f\in H_\omega}\frac{\|f-L_n(f)\|}{\omega(\frac\pi{n+1})}=1\quad(n=0,1,2,\dots), $$
where $\inf\limits_{L_n\in Z_n}$ is taken over all linear polynomial approximation methods of degree not higher than $n$ and $\sup\limits_\omega{}^*$ over all convex moduli of continuity $\omega(\delta)$.