Past exam of the approximation theory and finite element methods course of the University of Oxford / 2007 / A / 2 / d / Solution

The polynomial $T_6$ isSubtract its monic version from $x^6$:This has degree four and hence belongs to the usual space $P_5$. The error is $T_6/32$, with norm $1/32$ and alternating extrema at the seven points $\cos (j\pi /6)$, $j=0,\dots ,6$. Part (a), followed by uniqueness, provesand that this is the unique best approximation of degree at most five.

If “degree five” is taken literally to exclude degree four, no best approximation exists in that nonclosed class: $p+ϵx^5$ approaches the error $1/32$ as $ϵ\to 0$, but uniqueness in $P_5$ excludes attainment by any degree-five polynomial. This is why the standard at-most-degree convention matters here.