 Bernstein's lethargy theorem

Bernstein's lethargy theorem says that best approximation errors in nested approximation spaces can tend to zero arbitrarily slowly. For algebraic polynomials on $[-1,1]$, given any decreasing positive sequence ${\delta }_n\to 0$, there is a continuous function $f$ whose best degree-$n$ uniform-approximation error is at least ${\delta }_n$ for every $n$.