We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$. By considering the function:
$$\begin{aligned} f : \mathbb{R} &\rightarrow \mathbb{R} \\ y &\mapsto P(\cos y), \end{aligned}$$
verify that for all $x \in [-1,1]$,
$$P(x) = \|P\|_I \cos\left((n+m)\operatorname{Arccos} x\right).$$