Bernstein's lethargy theorem
= Bernstein's lethargy theorem
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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 $\delta_n\to0$, there is a continuous function $f$ whose best degree-$n$ uniform-approximation error is at least $\delta_n$ for every $n$.