We choose an even polynomial in $B_{N}$, denoted $R_{N}$, which has the factorisation $$R_{N}(X) = \prod_{j=1}^{r} \frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}} \prod_{k=1}^{s} \frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}} \prod_{\ell=1}^{t} \frac{X^{2} - w_{\ell}^{2}}{1 - w_{\ell}^{2}} \cdot \frac{X^{2} - \overline{w_{\ell}}^{2}}{1 - \overline{w_{\ell}}^{2}}.$$ After replacing all $\rho_k$ by zeros we obtained $S_N$. Similarly, in the list of $c_{j}$, we decide to replace those that do not belong to $[-1,1]$ by zeros. We thus replace the corresponding factors of $S_{N}$, $$\frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}}$$ by factors $X^{2}$. We thus obtain a new polynomial $T_{N}$. Show that $0 \leqslant T_{N}(x) \leqslant S_{N}(x)$ for all $x \in [-1,1]$, then that $T_{N} \in B_{N}$.
We choose an even polynomial in $B_{N}$, denoted $R_{N}$, which has the factorisation
$$R_{N}(X) = \prod_{j=1}^{r} \frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}} \prod_{k=1}^{s} \frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}} \prod_{\ell=1}^{t} \frac{X^{2} - w_{\ell}^{2}}{1 - w_{\ell}^{2}} \cdot \frac{X^{2} - \overline{w_{\ell}}^{2}}{1 - \overline{w_{\ell}}^{2}}.$$
After replacing all $\rho_k$ by zeros we obtained $S_N$.
Similarly, in the list of $c_{j}$, we decide to replace those that do not belong to $[-1,1]$ by zeros. We thus replace the corresponding factors of $S_{N}$,
$$\frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}}$$
by factors $X^{2}$. We thus obtain a new polynomial $T_{N}$.
Show that $0 \leqslant T_{N}(x) \leqslant S_{N}(x)$ for all $x \in [-1,1]$, then that $T_{N} \in B_{N}$.