Solution
= Solution
\b[True.] First choose polynomials $Q_n\to f$ uniformly by the <Weierstrass approximation theorem>. Let $L_1,\ldots,L_m$ be the <Lagrange interpolation polynomials> for the nodes $x_1,\ldots,x_m$, so $L_j(x_i)=\delta_{ij}$. Set
$$
P_n(x)=Q_n(x)+\sum_{j=1}^m\bigl(f(x_j)-Q_n(x_j)\bigr)L_j(x).
$$
Then $P_n(x_j)=f(x_j)$ for every $j$, while
$$
\|P_n-f\|_\infty
\leq\|Q_n-f\|_\infty
\left(1+\sum_{j=1}^m\|L_j\|_\infty\right)\longrightarrow0.
$$