Strict polynomial error decrease from nonvanishing derivatives ($E_{n-1}(f)>E_n(f)$)

If $n\ge 1$, $f$ is real and $C^n$ on an interval, and $f^{(n)}$ is nowhere zero there, the displayed inequality holds. If the errors were equal, a best degree-at-most-$n-1$ polynomial would also be best at degree $n$. The Chebyshev alternation theorem and intermediate value theorem would give $n+1$ zeros of the error. Applying Rolle's theorem $n$ times would force a zero of $f^{(n)}$, a contradiction. The zero-error case already contradicts $f^{(n)}\ne 0$. In particular this proves strict improvement for the exponential function at every positive degree step.