Estimation of trignometric polynomial and Lipschitz estimation

Question

Let $\mathcal{T}_n$ denote the linear space of trigonometric polynomials of degree up to $n$ and $$E_n(f)=\inf_{P\in\mathcal{T}_n} \|f-P\|_2=\|f-S_Nf\|_2=\left(\sum_{|k|>n}|\widehat{f}(k)|^2\right)^{\frac{1}{2}}$$ Let $0<\alpha<1$. Show that $E_n(f)\leq Cn^{-\alpha}$ for some constant $C$ iff $f\in \mathrm{Lip}_{\alpha, L^2}([0,1])$, that is $$\sup_{0<\delta<1}\sup_{|h|<\delta}\frac{\|f-f(\cdot -h)\|_2}{\delta^\alpha}<\infty.$$ The hint is "only if" part follows similarly by dyadic splitting of the frequencies. How do we show both parts, I tried various part and did not work out well.