We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. Let $(Q_1, R_1)$ be an extremal pair such that $\left\|Q_1\right\|_I = \left|Q_1(-1)\right|$ and $\left\|R_1\right\|_I = \left|R_1(1)\right|$. Let $n_1$ and $m_1$ be the degrees of $Q_1$ and $R_1$ respectively. We set $Q_2 = X^{n-n_1} Q_1$ and $R_2 = X^{m-m_1} R_1$. Show that $(Q_2, R_2)$ is a good extremal pair, i.e., $Q_2$ and $R_2$ are monic of degrees $n$ and $m$ respectively, $\frac{\|Q_2\|_I \|R_2\|_I}{\|Q_2 R_2\|_I} = C_{n,m}$, and $\|Q_2\|_I = |Q_2(-1)|$, $\|R_2\|_I = |R_2(1)|$.
We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. Let $(Q_1, R_1)$ be an extremal pair such that $\left\|Q_1\right\|_I = \left|Q_1(-1)\right|$ and $\left\|R_1\right\|_I = \left|R_1(1)\right|$. Let $n_1$ and $m_1$ be the degrees of $Q_1$ and $R_1$ respectively. We set $Q_2 = X^{n-n_1} Q_1$ and $R_2 = X^{m-m_1} R_1$.
Show that $(Q_2, R_2)$ is a good extremal pair, i.e., $Q_2$ and $R_2$ are monic of degrees $n$ and $m$ respectively, $\frac{\|Q_2\|_I \|R_2\|_I}{\|Q_2 R_2\|_I} = C_{n,m}$, and $\|Q_2\|_I = |Q_2(-1)|$, $\|R_2\|_I = |R_2(1)|$.