= Strict decrease of best polynomial approximation under a derivative sign
{title2=$f^{(n+1)}>0\Longrightarrow E_n(f)>E_{n+1}(f)$}
For real $f\in C^{n+1}[0,1]$ with positive $(n+1)$st <derivative>, equality of consecutive <best uniform approximation> errors would make a best degree-at-most-$n$ <polynomial> also best at degree $n+1$. The <Chebyshev alternation theorem> then supplies $n+3$ alternating extrema of its nonzero error, hence $n+2$ distinct zeros. Repeated <Rolle's theorem> forces a zero of the $(n+1)$st derivative of that error, contradicting the assumed sign. The same conclusion holds for a strictly negative derivative by replacing $f$ with $-f$.
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