Explicit Chebyshev construction for Bernstein lethargy ($E_n(f_{ϵ})\ge {ϵ}_n$)

For a strictly decreasing positive sequence ${ϵ}_n\to 0$, put $a_0={ϵ}_0-{ϵ}_1$ and $a_k={ϵ}_{3^{k-1}}-{ϵ}_{3^k}$ for $k\ge 1$. These are positive with sum ${ϵ}_0$. The positive lacunary Chebyshev series $f_{ϵ}={\sum }_k{a}_k{T}_{3^k}$ has error ${ϵ}_0$ at degree zero and error ${ϵ}_{3^{K-1}}\ge {ϵ}_n$ whenever $3^{K-1}\le n<3^K$. This proves the lower-bound form of Bernstein's lethargy theorem with an explicit continuous function.