equality (\[keyeq\]), there exists a number $N>0$ such that for every $n \geq N$ the function $S_{1/\rho_n, \mu_n,n}(x)>0$ on $[-1,\,1]$ particularly at the nodes. According to Lemma \[mainsec\], the coefficients $w_{j,n}$, $j=1, \ldots, n$, are also positive. This completes the proof.

\[lemaclave\] Consider the varying measure $\mbox{d}\, \mu_n(x)=\mbox{d} \, \mu(x)/q_{m(n)}(x)$ and their orthogonal polynomials $\displaystyle q_{\mu_n,n}(x)=\prod_{j=1}^n\left(x-x_{j,n}\right)$ and $\displaystyle q_{\mu_n,n-1}(x)=\prod_{j=1}^{n-1}\left(x-x_{j,n-1}\right)$, $n\in \mathbb{N}$. Let $\displaystyle {\bf y}=\left\{{\bf y}_n=\left(y_{1,n},\ldots, y_{n,n}\right)\right\}_{n\in \mathbb{N}}$ be a scheme of nodes such that for each $n\in \mathbb{N}$ $$\label{entrelazamiento}
-1<y_{1,n}<x_{1,n-1}<y_{2,n}<\cdots < x_{n-1,n-1}<y_{n,n} <1.$$ Assume that the polynomials $\displaystyle P_{n}(x)=\prod_{j=1}^n \left(x-y_{j,n}\right), \quad n\in \mathbb{N}$ satisfy $$\label{extructura}
\lim_{n \to \infty}\frac{1}{\left|\left|q_{\mu_n, n-1}\right|\right|^2_{2, \mu_n}}\left[\frac{(q_{\mu_n,n}-P_n)q_{\mu_n,n-1}}{q_{m(n)}^2}\right]^{\prime}=0, \quad \mbox{on} \quad [-1,\,1].$$ Then ${\bf y}$ is convergent.

From Lemma \[interwendroff\] we ensure the existence of a weight function $\rho_n$ such that the polynomials $q_{\tau_n,n-1}$ and $P_{n}$ belong to the family of orthogonal polynomials corresponding to the measure $\rho_n(x)\mbox{d} \mu (x)/q_{m(n)}(x)$. Let us analyze the function $$\left|1-\frac{\left|\left| P_{n}\right|\right|^2_{2, \rho_n d \mu/q_{m(n)}}}{q_{m(n)}(x) \left|\left|q_{\mu_n, n-1}\right|\right|^2_{2, \mu_n}}S_{1/\rho_n, \rho_n d \mu/q_{m(n),n} }(x)\right|$$ $$=\frac{1}{q_{m(n)}(x)}\left|S_{q_{m(n)}, \mu_n,n}(x) -\frac{\left|\left| \widetilde{P}_{n}\right|\right|^2_{2, \rho_n d \mu/q_{m(n)}}}{\left|\left|q_{\mu_n, n-1}\right|\right|^2_{2, \mu_n}}S_{1/\rho_n, \rho_n d \mu/q_{m(n),n} }(x)\right|$$ We have used that $\displaystyle S_{q_{m(n)}, \mu_n,n}\equiv q_{m(n)}$, hence we need to show that $$\lim_{n\to 0}\frac{1}{q_{m(n)}(x)}\left|S_{1/\rho_n, \mu_n,n}(x)-\frac{\left|\left|\widetilde{P}_{n}\right|\right|^2_{2, \rho_n d \mu/q_{m(n)}}}{\left|\left|q_{\mu_n, n-1}\right|\right|^2_{2, \mu_n}}S_{1/\rho_n, \rho_n d \mu/q_{m(n),n} }(x)\right|=0.$$ Applying 