= Alternating-reference error bound
{title2=$|h|\leq E_n(f)\leq\|f-q\|_\infty$}
Let a <polynomial> $q$ of degree at most $n$ have errors $f(x_j)-q(x_j)=(-1)^jh$ at $n+2$ ordered distinct points. Then $|h|$ bounds below the error $E_n(f)$ of the <best uniform approximation>. Otherwise an improved <polynomial> $p$ with error strictly below $|h|$ would make $p-q$ alternate strictly in sign at all those points, giving at least $n+1$ distinct <roots of a polynomial> by the <intermediate value theorem>. Its degree is at most $n$, a contradiction. Moreover, if $q$ is not optimal, $|h|<E_n(f)$. In the equality case, an optimal $p$ gives weak alternating signs of $p-q$; its vanishing order-$(n+1)$ <divided difference> has same-sign summands, forcing every nodal value to vanish and hence $p=q$.
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