Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 75 / 3 / b / Solution

Let $p$ be the best uniform approximation to the exponential function from the degree-at-most-$n-1$ polynomial space. Suppose that $E_{n-1}(e^x)=E_n(e^x)$. Then the same $p$ is best in the larger space. Its error $g(x)=e^x-p(x)$ is nonzero, since the exponential is not a polynomial. The Chebyshev alternation theorem supplies $n+2$ ordered points where $g$ has alternating signs. By the intermediate value theorem, $g$ has at least one zero in each of the $n+1$ intervening intervals, hence at least $n+1$ distinct zeros.

Applying Rolle's theorem successively $n$ times forces a zero of $g^{(n)}$. But $\mathrm{deg}p\le n-1$, soeverywhere. This is a contradiction. Together with the automatic non-increase of best errors, it provesThe argument is an instance of strict polynomial error decrease from nonvanishing derivatives.